Instabilities, Chaos And Turbulence: An Introduction To Nonlinear Dynamics And Complex Systems

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Instabilities, Chaos and Turbulence An Introduction t o Nonlinear Dynamics and Complex Systems

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Instabilities, Chaos and Turbulence An Introduction t o Nonlinear Dynamics and Complex Systems

Paul Manneville Ecole Polytechnique, France

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE

Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

INSTABILITIES, CHAOS AND TURBULENCE An Introduction to Nonlinear Dynamics and Complex Systems Copyright 0 2004 by Imperial College Press

All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-483-3 ISBN 1-86094-491-4 (pbk)

Printed in Singapore by World Scientific Printers (S) Pie Lid

To Jacqueline, Jean-Baptiste, SCbastien, Alexis, and Claire.


Preface

During the last quarter of the Twentieth Century, the study of nonlinear and complex systems has experienced unprecedented development. In the following pages, we present a first approach t o standard results and recent advances in this field, applied to phenomena at our scale in our surrounding world. The pretext chosen is mainly instabilities in out-of-equilibrium systems, and more specifically the transition to turbulence in flowing fluids. It should however be clear that the methods used are of fully general use. The book is based on lecture notes for a short optional course given to second-year students in an engineering school, Ecole Nationale Supkrieure de Techniques Avanckes, in Paris. This school is devoted to the training of high level engineers in fields including applied mathematics, mechanics and hydrodynamics, electronics,. . . , oceanography, and management. At the time of the course, students have not yet chosen their specialty, so the course has t o be sufficiently general and without too specific requisites. Accordingly, the book should be of interest to nearly any science-oriented undergraduate student and, potentially, to everybody wanting t o learn about recent advances in the field of applied nonlinear dynamics. Technicalities are not completely avoided but they are explained as simply as possible using heuristic arguments and specific worked examples, while openings on different topics can be gained by solving exercises at the end of each chapter using the same methods as those explained in the text. At first, the problem of chaos that one has to face very early in this field may seem abstract and difficult. Even if the treatment of examples is not complete, the reader should get a concrete and operational mastery of concepts and techniques to be used from them. As far as the difficulty is concerned, our aim is to transmit the knowledge rather informally and without full mathematical rigor. With respect t o mathematics and physics, vii

...

Vlll

Instabilities, Chaos and Turbulence

only basic understanding is required, at the level of what is currently known after one or two years of undergraduate training. In mathematics, this does not go further than elementary algebraic calculus, basic notions of linear algebra and ordinary differential calculus. As far as physics is concerned, it should suffice to follow one’s intuition and to admit a few fundamental equations without discussion. So, adaptation of the approach to any other field of interest should thus be envisaged without excessive anxiety. A first brief chapter situates the context of the study, that of evolutionary problems involving a specific independent variable called time, distinguishing discrete systems governed by finite sets of ordinary differential equations and continuous media described by partial differential equations. It serves as an introduction to the rest of the book, explaining in particular that continuous media driven out of equilibrium may experience instabilities inducing structures that further break down, leading to turbulence. The second chapter is devoted to a preliminary study of dynamical systems with a small number of degrees of freedom. The archetype of such systems is the oscillator which serves to introduce the first manifestations of nonlinear effects, e.g. the occurrence of self-oscillations or the relation between amplitude and frequency. The way to complex behavior is then apparently left a t a too early stage, before the occurrence of chaos. In Chapter 3 we indeed turn to a specific but particularly simple and intuitive physical problem, the stability of a fluid layer heated from below and entering a convection regime. The first part of the chapter is devoted to the analysis of the instability mechanism and an approximate determination of the threshold. In the second part, a description is given of the “death” of the so-formed dissipative structure (pattern of convection rolls) and of the steps toward turbulence. After this detour, we come back to the mathematics of the transition within the dynamical systems framework. A preliminary step of the reduction of the dynamics has to be performed, resting on the distinction between driving and enslaved modes, and on elimination of the latter modes. The emergence of complexity is then analyzed as a result of the increase of the number of driving modes. This is done in two steps. The first one constitutes the last part of Chapter 4 where we introduce scenarios of transition to temporal chaos and present some of the tools used to identify it and measure its amount from an empirical point of view. However, as already observed when studying convection, confinement effects play an important role, and all that precedes relates to the case when they are strong enough t o freeze the spatial structure of the modes. Otherwise, scenarios relevant

Preface

ix

to extended systems, the second step, involve large scale modulations and spatio-temporal chaos, both introduced in Chapter 5. In common language, the third element of the title, turbulence, refers to the irregular, highly fluctuating, behavior of most of the flows surrounding us (the opposite situation of so-called laminar flows is rather exceptional). This problem is tackled in two steps. Instability and transition of open flows is examined in Chapter 6 . By contrast with systems considered in Chapter 3, where the fluid remained confined t o an enclosure, it now circulates from upstream to downstream, the consequences of which will be sketchily discussed. In Chapter 7, we consider developed turbulence, again along two paths. First we analyze the different scales, from the largest where energy is injected eddies by instability mechanisms t o the smallest where it is consumed via viscous dissipation. In a second instance, we turn to the statistical problem of predicting the average properties of a given turbulence flow, of utmost practical interest for an engineer. Chapter 8 recapitulates the results and opens the perspective toward a complex dynamical system of contemporary interest, the climate of the Earth, and the problem of understanding/modeling its past and present trends. A first appendix is devoted to a summary of linear algebra results that are useful throughout the book. As far as the understanding of nonlinear phenomena is concerned, the recourse to computers has been of considerable help a t different levels. This is the reason why a second appendix, introducing hands-on computer sessions, is devoted to elementary methods that can be developed, with sufficient common sense and no superfluous specialization, to extract useful information from numerical simuIations of simple, even simplistic, but well-designed generic models of nonlinear dynamics and pattern formation. The course is completed by laboratory sessions on topics, the theory of which is considered in some exercises. Palaiseau, June 2004.

X


Not ations. Here are some indications about the conventions that, with as few exceptions as possible, we will be using throughout. First, upper-case bold letters will denote points in the spaces that will be considered, e.g. X E X, X will most often be a real vector space Rd. Components will be XI, X2, ..., Xd, collectively denoted {Xi}. We will also possibly use the bold and normal upper-case Greek letters in the same context. For operators working in these spaces, we will use the bold ‘cal’ TEX font, e.g. 3, and for their components the normal ‘cal’ font with indices, e.g. Fi,hence Y = 3 ( X ) and y3 = Fj({Xi}), i , j = 1,.. . , d . In the case of a linear operator, e.g. C, we will rather write Y = CX. In general, we will not distinguish the operator from the matrix that represents it in a given basis, C will denote, with little ambiguity, either the operator or the matrix with elements l j j l . The above equation would then read in developed form: y3 = C i l j i X i . Some elements of linear algebra are recalled in Appendix A. With respect to differentiation, for the ordinary derivative with respect to some variable U we will use d(...)I dU. With respect t o time t, we usually prefer dots surmounting the variable, one for each differentiation order: dX/dt 3 X and d2X/dt2 X . A short-hand notation for the partial derivative with respect to variable X, a ( . . . ) / a X , will be ax(...). Points in physical space will usually be noted x with coordinates x,y, z , corresponding unit vectors % , g , i ,and partial derivatives d,, 8,. In the same way, a velocity vector will be noted V or v with components V,, V,, V, or v,, v,, v,. On some occasions we will rather take xj, vj, and , j = 1 , 2 , 3 allowing us to use the Einstein convention of implicit sum over repeated indices. The complex conjugate of a complex number 2 = 2, +iZi, Re(2)= Zr, Zm(Z)= Zi, will be noted as Z * . For quantities that play the role of control parameters, we will not follow the convention used in fluid mechanics to take two letters, e.g. ‘Re’ for the Reynolds number of a given flow, ‘Ra’ for the Rayleigh number in convection, etc., but a simple ordinary letter, upper or lower case, most usually R or T , to stress the fact that control parameters are ordinary variables (ambiguities will be raised on a case-by-case basis). Numerical illustrations have most often been obtained using MATLAB. We will occasionally use some of its conventions.

a,,

a,,

Contents

Preface

vii

1. Introduction and Overview

1

1.1 Dynamical Systems as a Context . . . . . . . . . . . . . . . 1.2 Continuous Media as a Subject . . . . . . . . . . . . . . . . 1.3 From Simple t o Complex . . . . . . . . . . . . . . . . . . . 1.3.1 Thermal convection: the instability mechanism . . . 1.3.2 Nonlinear convection and dynamical systems . . . . . 1.3.3 Stability and instability of open flows . . . . . . . . .

2 5 9 10 12 15

..

17 20

1.3.4 Beyond the transition: fully developed turbulence 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Exercises

............................

2. First Steps in Nonlinear Dynamics 2.1 From Oscillators to Dynamical Systems . . . . . . . . . . . 2.1.1 First definitions . . . . . . . . . . . . . . . . . . . . . 2.1.2 Formalism of analytical mechanics . . . . . . . . . . 2.1.3 Gradient systems . . . . . . . . . . . . . . . . . . . . 2.2 Stability and Linear Dynamics . . . . . . . . . . . . . . . . 2.2.1 Formulation of the linear stability problem . . . . . . 2.2.2 Two-dimensional linear systems . . . . . . . . . . . . 2.2.3 Stability of a time-independent regime . . . . . . . . 2.3 Two-dimensional Nonlinear Systems . . . . . . . . . . . . . 2.3.1 Two examples of oscillators . . . . . . . . . . . . . . 2.3.2 Amplitude and phase of nonlinear oscillators . . . . . 2.4 What Next? 2.5 Exercises .

........................... ........................... xi

20 25 25 25 32 33 35 36 37 41 43 44 50 60 61

xii

Instabilities. Chaos and firbulence

3. Life and Death of Dissipative Structures

71

71 3.1 Emergence of Dissipative Structures . . . . . . . . . . . . . 3.1.1 Qualitative analysis of the instability mechanism . . 71 3.1.2 Simplified model . . . . . . . . . . . . . . . . . . . . 73 3.1.3 Normal mode analysis, general perspective . . . . . . 75 3.1.4 Back to the model . . . . . . . . . . . . . . . . . . . 78 3.1.5 Vicinity of the threshold: linear stage . . . . . . . . . 82 3.1.6 Classification of unstable modes . . . . . . . . . . . . 84 3.2 Disintegration of Dissipative Structures . . . . . . . . . . . 86 3.2.1 Simplified model of nonlinear convection . . . . . . . 86 3.2.2 Transition to turbulence of convection cells . . . . . 90 3.2.3 Transition toward chaos in confined systems . . . . . 94 3.2.4 Dynamics of “textures” in extended systems . . . . . 97 3.2.5 Turbulent convection . . . . . . . . . . . . . . . . . . 101 3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 . Nonlinear Dynamics: from Simple to Complex 4.1 Reduction of the Number of Degrees of Freedom . . . . . . 4.1.1 Role of aspect ratios . . . . . . . . . . . . . . . . . . 4.1.2 Low dimensional effective dynamics . . . . . . . . . . 4.1.3 Center manifolds and normal forms . . . . . . . . . . 4.2 Transition to Chaos . . . . . . . . . . . . . . . . . . . . . . 4.2.1 First steps: time-independent and periodic regimes . 4.2.2 Quasi-periodicity and resonances . . . . . . . . . . . 4.2.3 Quasi-periodicity and lockings . . . . . . . . . . . . . 4.3 Characterization of Chaotic Regimes . . . . . . . . . . . . . 4.3.1 Instability of trajectories and Lyapunov exponents . 4.3.2 Fractal aspects . . . . . . . . . . . . . . . . . . . . . 4.4 Empirical Approach of Chaotic Systems . . . . . . . . . . . 4.4.1 Standard analysis by means of Fourier transform . . 4.4.2 Reconstruction by the method of delays . . . . . . . 4.4.3 Sampling frequency and embedding dimension . . . . 4.4.4 Application . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 116 116 118 121 125 126 130 136 147 148 150 154 156 157 160 165 168 168

Contents

5. Nonlinear Dynamics of Patterns 5.1 Quasi-one-dimensional Cellular Structures . . . . . . . . . . 5.1.1 Steady states . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Amplitude equation . . . . . . . . . . . . . . . . . . . 5.2 Dissipative Crystals . . . . . . . . . . . . . . . . . . . . . . 5.3 Short Term Selection of Patterns . . . . . . . . . . . . . . . 5.4 Modulations and Envelope Equations . . . . . . . . . . . . . 5.4.1 Quasi-one-dimensional cellular patterns . . . . . . . . 5.4.2 2D modulations of quasi-1D cellular patterns . . . . 5.4.3 Quasi-two-dimensional cellular patterns . . . . . . . 5.4.4 Oscillatory patterns and dissipative waves . . . . . . 5.4.5 Universal long-wavelength instabilities . . . . . . . . 5.5 What Lies Beyond? . . . . . . . . . . . . . . . . . . . . . . . 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Open Flows: Instability and Transition 6.1 Base Flow Profiles . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Strictly one-dimensional flows . . . . . . . . . . . . . 6.1.2 More general velocity profiles . . . . . . . . . . . . . 6.1.3 Extension to arbitrary profiles . . . . . . . . . . . . . 6.2 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 General framework . . . . . . . . . . . . . . . . . . . 6.2.2 Inviscid flows . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Viscous flows . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Instability and downstream transport . . . . . . . . . 6.3 Transition t o Turbulence . . . . . . . . . . . . . . . . . . . . 6.3.1 Nonlinear development of instabilities . . . . . . . . 6.3.2 Inviscidly unstable flows . . . . . . . . . . . . . . . . 6.3.3 Inviscidly stable flows . . . . . . . . . . . . . . . . . 6.3.4 Turbulent spots and intermittency . . . . . . . . . . 6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Developed Turbulence

...

Xlll

181 182 182 185 187 190 191 191 193 195 197 199 204 205 211

213 213 215 218 219 219 223 231 234 241 241 244 249 253 265 271

7.1 Scales in Developed Turbulence . . . . . . . . . . . . . . . . 272 7.1.1 Production scale . . . . . . . . . . . . . . . . . . . . 272 7.1.2 Inertial scales and Kolmogorov spectrum . . . . . . . 274 7.1.3 Dissipation scales . . . . . . . . . . . . . . . . . . . . 276 7.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 278

Instabilities. Chaos and Turbdence

xiv

7.2 Mean Flow and Fluctuations . . . . . . . . . . . . . . . . . 7.2.1 Statistical approach . . . . . . . . . . . . . . . . . . . 7.2.2 Reynolds averaged Navier-Stokes (RANS) equations 7.2.3 Energy exchanges in a turbulent flow . . . . . . . . . 7.3 Mean Flow and Effective Diffusion . . . . . . . . . . . . . . 7.3.1 Mixing length and eddy viscosity . . . . . . . . . . . 7.3.2 Application to the determination of the mean flow . 7.4 Beyond the Elementary Approach . . . . . . . . . . . . . . 7.4.1 Turbulence modeling . . . . . . . . . . . . . . . . . . 7.4.2 Large eddy simulations . . . . . . . . . . . . . . . . . 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Summary and Perspectives

8.1 8.2 8.3 8.4

303

Dynamics. Stability. and Chaos . . . . . . . . . . . . . . . . 304 Continuous Media. Instabilities. and Turbulence . . . . . . 308 Approach to a Complex System: the Earth’s Climate . . . . 312 Exercise: Ice ages as catastrophes . . . . . . . . . . . . . . . 328

Appendix A

A.l A.2 A.3 A.4

279 279 280 283 284 284 286 290 292 294 299

Linear Algebra

Vector Spaces. Bases. and Linear Operators . . . . . . . . . Structure of a Linear Operator . . . . . . . . . . . . . . . . Metric Properties of Linear Operators . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix B Numerical Approach

B.l Treatment of the Time Dependence . . . . . . . . . . . . . . B.2 Treatment of Space Dependence in PDEs . . . . . . . . . . B.2.1 Finite difference methods . . . . . . . . . . . . . . . B.2.2 Spectral methods . . . . . . . . . . . . . . . . . . . . B.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.1 ODEs 1: Forced pendulum . . . . . . . . . . . . . . . B.4.2 ODEs 2: Lorenz model . . . . . . . . . . . . . . . . . B.4.3 ODEs 3: Rossler and Chua models . . . . . . . . . . B.4.4 PDEs 1: SH model, finite differences . . . . . . . . . B.4.5 PDEs 2: SH model, pseudo-spectral method . . . . . Bibliography Index

331

331 334 343 348 35 1 352 358 358 363 369 371 371 373 376 378 380 383 387

Chapter 1

Introduction and Overview

In a linear world, the effects being always proportional to their causes, everything would be “simple” since the tools that would allow us to represent it as a superposition of elementary states are rather well mastered. Unfortunately the world is “complex.” As a matter of fact, we often observe effects which saturate in spite of an increase of their causes, or which go in different and somehow unexpected ways. All this is ascribed to nonlinearities. Of course, if we succeed in determining the state of a nonlinear system, we immediately try to go back to a problem that we know how t o handle by linearizing the dynamics around it and treating small departures from it by a perturbative approach, and next to reproduce this scheme as far as possible in order t o reach other fully nonlinear states, eventually “far from” the initial one. Most recently, our science of intrinsically nonlinear phenomena has made much progress (even though it rests in part on the use of linear tools) and investing this knowledge further in applications is of primary interest. The aim of these lecture notes is therefore t o introduce the reader t o this breakthrough by taking the problem of macroscopic instabilities as a pretext. In thermodynamic systems close to equilibrium the response to excitations is essentially linear, i. e. proportional to the (sufficiently small) amplitude of the applied stress with a proportionality factor called a susceptibility. As a consequence of linearity, the regime that develops is unique. When the applied stress increases, the system is driven farther from equilibrium and nonlinearities can no longer be neglected. This opens the possibility of bifurcations towards different regimes that can coexist and compete. As a result of such “catastrophes,” losing reference to the initial state the dynamics becomes increasingly complicated, typically from “laminar” t o “turbulent flow.” 1

2

1.1

Instabilities, Chaos and Twbdence

Dynamical Systems as a Context

In order to study the causes of the complexity induced by nonlinearities, we will concentrate our attention on problems defined in terms of the evolution of a set of state variables functions of a single independent variable called t i m e (the general concept of dynamical system). Mechanical systems in the ordinary sense obviously belong to this class. Their archetype is the oscillator, and in its simplest linear expression, the harmonic oscillato9 (Figure 1.1, left)

m X = F = -kX = 0 ,

(1.1)

where X measures the departure from the equilibrium length of the spring, m is the mass attached to its end, and F is the restoring force, here taken proportional to X via the stiffness coefficient k. As a result the oscillation period is independent of the amplitude: T = TO= 2.rr/wo, with w; = k / m . At this stage’ nonlinearities can introduce themselves in two ways. First it is easy to imagine that the stiffness coefficient may not be a constant but rather a function of X itself. Assuming k = ko(1 ax2)leads to what is known as the D u f i n g oscillator. Another possibility comes from the existence of mechanical constraints. The ideal rigid pendulum, a mass m at the end of a weightless rod of length 1 and revolving around a horizontal axis in the gravity field, is a good example (Figure 1.1, right). Parameterizing its position by the angle 8 it forms with the vertical axis, one obtains:

+

J e + m g l sin 8 = 0 ,

(1.2)

where J = m12 is the moment of inertia and -mgl sin 8 the torque exerted by its weight. The nonlinearity built in the sine function is a consequence of the topological constraint keeping the mass rigidly at a constant distance from the rotation axis and plays a role in case of large deviations. As an outcome, several equilibrium positions exist. The ‘down’ position is a stable equilibrium point, with small amplitude oscillations around it governed by the same equation as the harmonic oscillator. The ‘up’ position is unstable and the pendulum departs from it upon perturbations of any amplitude, even infinitesimal. Furthermore low energy states oscillate around the ‘down’ position with a period depending on the energy, while a t larger energies, the system rotates always in the same direction, accelerating and slowing down periodically as it passes through the ‘down’ and ‘up’ lAs indicated at the end of the preface, when the independent variable is time, differentiation is denoted by dots on top of dependent variables.

3


Fig. 1.1 Left: Linear spring. Right: Rigid pendulum.

positions, respectively. The behavior of nonlinear oscillators will be further considered in Chapter 2. Such mechanical examples can serve us to introduce a geometrical description of the dynamics in a space called the phase space, here with coordinates the positions X or 0 and the momenta mX or J d , and to account for the dynamics in terms of trajectories in that space. In practice, any system involving a finite set of dependent variables serving to characterize its state, its state variables, governed by differential equations, belongs to the class we consider here. Examples are intensities and potential differences in an electrical circuit (Chapter 2, §2.3.1.2),concentrations of reactants in chemistry (Exercise 1.5.1), population densities in ecology (Exercise 1.5.2),.. . In analytical mechanics, a pair ‘position conjugate momentum’ is called a degree of freedom. In a more general context, what is called a degree of freedom is simply a state variable. In general, it is advantageous to write down the dynamical equations as a system of first order differential equations

+

X j

= 3 j ( { X i ( t ) i; = 1 , . . . , d } , t ) ,

j = 1 , . . . ,d ,

where the integer d is, in this context, called the dimension of the system.2 Using the notations X and F for the sets {Xj,j= 1 . . . d } , and {3j,j = 1 . . . d}, we have thus:

x = 3 ( X ,t ) .

(1.3)

2Accordingly, the dimension of the phase space is twice the number of degrees of freedom in analytical mechanics whereas otherwise it is just the number of state variables. In the following we will try to avoid ambiguities arising from this terminology. See later, Chapter 2, 52.1.2.


4

For a system such as (1.3) time is a continuous variable. By contrast, a discrete-time system is defined as an iteration that we can write in the form Xj,k+l

= @j ( { X i , k ; i = 1 , . . . , d } ,k),

j = 1 , . .. , d .

or, formally:

For such a system, time is the discrete index k that serves to monitor the evolution. While appearing as a topic in itself in mathematics, the study of discrete-time dynamical systems turns out to be essential in physics and engineering owing to their occurrence as a result of the stroboscopic analysis of periodically forced continuous-time dynamical systems, or of a PoincarC section in self-oscillating systems. They can also introduce themselves as the outcome of modeling effort of specific phenomena, e.g. seasonal counts in population dynamics. All this will be considered in detail in Chapter 4. Systems considered here all carry out the intuitive concept of determini s m which imply the prediction of a state at time t or lc beyond some time t o or ICo at which some initial condition is specified. This initial value problem presents itself formally or explicitly as a computational problem, directly if it is defined as an iteration (1.4) or indirectly through some numerical approximation for a continuous-time system (1.3), e.g. Euler extrapolation,

since analytical integration is rarely possible. See Appendix B, fjB.1, for an introduction. Resuming the geometrical perspective introduced above, we are now interested in the properties of trajectories followed by the system in its phase space. This study rapidly points to the key role played by the concept of stability taken in the broad sense of resistance t o perturbations. In practice, this vague definition refers to two different viewpoints: The first one, rather quantitative, applies to specific trajectories: the solutions found, which depend on initial conditions, have to withstand small perturbations imposed either at the start or during the subsequent evolution, owing to unavoidable disturbances, either intrinsic (thermodynamic fluctuations) or extrinsic (noise). From this first viewpoint, only stable solutions (that resist) are physically observable. Further, a sufficiently large set of experimental conditions should make them attainable.

1 . Introduction and Overview

5

The second viewpoint is mode qualitative. It refers to the very definition of the system itself: when compared to the real world, any abstract implementation is intrinsically blurred with numerous approximations and its control parameters are not determined with infinite precision. Accordingly, in order to be of help, the analytical model must be robust, i.e. its predictions must not be too sensitive to these different sources of inaccuracy. This property indeed fails at bifircation points where the system experiences qualitative changes of behavior. At such points one says that it is structurally unstable: the nature of the state depends sensitively on the perturbation. It can be easily understood that these two facets of the term ‘stability’ are equally important for the applications. They will be at the heart of the most abstract part of the course, first in Chapter 2 and next in Chapter 4 where we will give a more precise meaning t o the word ‘prediction’ when the considered system evolves chaotically, that is t o say in a way unpredictable in the long term in spite of short term determinism, due to an instability of trajectories that is the essence of chaos.

1.2

Continuous Media as a Subject

Dynamical systems considered up to now were endowed with a supposedly small number of dependent variables, thus living in spaces of low enough dimensions. Once the microscopic structure of matter has been recognized, in principle one should turn t o the study of systems made of a large number of components at the molecular scale, each with its own degrees of freedom in the mechanical sense. However a refined description of the microscopic configurations is generally useless (due to chaos at this scale) and our ignorance of dynamical details can be circumvented by adopting a statistical point of view from which the less probable (specific initial condition) evolves into the most probable (equilibrium compatible with conservation laws), founding the thermodynamic approach. In this framework, microscopic variables are replaced by average quantities such as energy, entropy, temperature or pressure. In practice, global thermodynamic equilibrium is not of much interest since it accounts for a world eaten by the worm of the equiprobability of microscopic states, and in some sense completely dead. On the contrary, an inhomogeneous world traversed by various fluxes making it alive (which by the way allows us to study it!) is much richer and more interesting. We

6


are indeed confronted to a large class of transport processes in media that are out of equilibrium on a macroscopic scale and there is a wide range of time and space scales for which the concept of continuous medium is appropriate, that is to say the description of a system according to which its state variables are functions of the position in physical (three-dimensional) space, X(x, t ) , governed by partial differential equations:

atx = S ( X ,vx,t ) .

(1.6)

From a mathematical perspective, such systems are infinite-dimensional since we need to specify the value of these variables at every point in space. On the other hand the validity of the description relies on the definition of a mesoscopic scale inbetween the microscopic (molecular) level and the (macroscopic) size of the system taken as a whole. The concept of material point and the assumption of local equilibrium make sense precisely on this intermediate scale, sufficiently small to be considered as infinitesimal but large enough to contain as many molecules as necessary for the laws of thermodynamic to be relevant. Equations such as (1.6) are in general derived from the macroscopic balance of extensive thermodynamic variables, say 2, that, in differential form, read

where pz is the density of 2, J, its flux, and C, a source term that cancels when 2 is a conserved quantity (momentum, energy). The simplest example is the Fourier heat equation governing thermal diffusion:

C&T = x V 2 T ,

(1.8)

governing the temperature field in a solid submitted to moderate gradients. It derives from the Fourier law

that links the heat flux to the temperature gradient through the thermal conductivity x. Quantity C in (1.8) is the specific heat defined through the thermodynamic relation 6U = C6T. Inserting this into the balance V . JQ = 0, where prr is the corequation for the internal energy atp, responding density obtained as the limit of bU/6V as the volume element 6V + 0, immediately yields (1.8).

+

7


The quantity K = x / C is the thermal diffusivity. Dimensionally, this coefficient is homogeneous to [LI2/[T], like the diffusivity D appearing in the Fick law governing molecular diffusion. In this context, an important quantity is the time scale T for diffusive relaxation over a typical space scale L, directly obtained from the dimensional relation above as r = C 2 / & . The relaxation of T is examined in the most elementary case in Exercise 1.5.3 which further illustrates a rare situation where one can find an analytic transform allowing the exact linearization of a parent nonlinear problem. Equation (1.8) is indeed linear provided that the specific heat C and the thermal conductivity x are constant, which is reasonable not to far from equilibrium. The Fourier law (1.9) is the prototype of a phenomenological relation linking a flux (the effect) to a gradient (the cause) at a linear level. In practice, the validity range of such linear law is generally rather wide, because constraints that we are able to apply to continuous media are usually weak when compared to molecular interactions. As counter-examples one could cite electronic systems containing active elements with threshold effects (diodes), or chemically reacting media, and a fortiori biological systems. However, in macroscopic media strong nonlinearities can in fact arise from global considerations. This is particularly the case in fluid systems where macroscopic flow deeply affects the transport properties, rendering a locally linear medium effectively nonlinear. So, in hydrodynamics the concept of the material p o i n t transforms itself into that of fluid particle, the position of which becomes a function of time:

It is most often useful to pass from this Lagrangian description, to an Eulerian approach in terms of velocity field:3 (vZ7vy,vZ)

($Z:M,

d

ZyM,

d ZZM)

.

The two descriptions are linked by the definition of the material derivative measuring the evolution of any physical quantity 2 attached to the fluid particle as it is followed during its motion: 2 = 2 (zM,yM,z M ,t ) . Mathematically, material differentiation is thus a total differentiation with respect to time, $2, that is: $2 = at2

+ a,z

&EM

+ ayz&yM + a,z

$ZM

= at2 + v

*

vz.

(1.10)

3Here we no longer denote time differentiation by dots but come back with explicit derivatives in order to avoid ambiguities.

8


This relation allows a simple expression of the balance equation in differential form provided that the flux of 2 is properly split into an irreversible diffusive part and a reversible part linked to the macroscopic motion: Jz

= Jz,diff

+ PZV

*

When studying the mechanism of natural convection in fluids originally at rest we will demonstrate the role of a global nonlinearity played by the advection term in spite of its apparent linearity in p a , but before considering this example, let us recapitulate the equations governing the motion of a simple Newtonian fluid, i.e. a single component fluid with a constant viscosity. The first one is the continuity equation accounting for the conservation of matter. In full generality it reads:

&p+V.J, =O, (no source term). Noticing that diffusion of matter within itself does not make sense, we get J, = pv which leads to & p + v . v p + p V .v

= &p+

p v .v = 0.

(1.11)

In the following we will consider incompressible flows characterized by & p = 0, so that the continuity equation more simply reads

v . v = 0.

(1.12)

On the other hand, for an incompressible fluid the compression viscosity drops out and there only remains a shear viscosity which, for an isotropic Newtonian fluid, relates the viscous stress tensor t o the rate of strain in a linear way: r.' a j - p(8iV.j

+ 8j.i)

7

where p is the dynamical viscosity. Once inserted into the momentum conservation equation, this relation leads t o the Navier-Stokes equation

p(&+v.V)v= -Vp+pV%.

(1.13)

The kinematic viscosity u = p / p is the transport coefficient accounting for the diffusive relaxation of velocity gradients (Stokes law). At the inviscid limit ( u -+ 0) the Navier-Stokes equation is called the Euler equation. The opposite limit where the viscous dissipation dominates is called the Stokes approximation. Finally if the heating generated by viscous friction


9

is negligible, energy conservation yields the heat equation (1.8) where &T is just replaced by ZT (at v . V)Tl hence:

+ &T + v . VT = ~ G V ’ T .

1.3

(1.14)

From Simple to Complex

In a narrow vicinity of thermodynamic equilibrium, the response of a continuous system is simply proportional to the strength of applied constraints and displays the same symmetries, stationary or periodic in time, uniform or periodic in space, for example. However, far from equilibrium, when nonlinearities can no longer be neglected] the system can bifurcate towards solutions that breaks some of these symmetries. The nature of the regime that develops depends on the value of control parameters measuring the relative intensity of the different contributions to the dynamics. In fluid mechanics, it will most often be the Reynolds number defined as R = U e / u where U and e represent a typical velocity and a length scale both characteristic of the flow under consideration while u, the kinematic viscosity introduced above, is a fluid’s property. In line with what has been said about the thermal diffusion time, it is enlightening to analyze the Reynolds number as the ratio of the viscous relaxation time over the length e, rv = e 2 / u ,to the advection time required to carry velocity fluctuations over the same distance, T, = e / U . When the Reynolds number R = rv/Ta is small, viscosity has time to wipe out inhomogeneities, whereas when it is large, thermodynamic dissipation is too slow and the mechanical contribution of advection dominates. The study of the stability of a given regime called the base state rests on the definition of perturbations, i.e. departures from this state, and their subsequent evolution. Figure 1.2 illustrate the general situation. Below a value of the control parameter R denoted R,,‘g’ for global, the base state is unconditionally stable: whatever the shape and the amplitude of the perturbation, it decays and the system returns to its base state (sufficient condition of stability). Above a second value R,, ‘cl for critical, the system is sensitive to at least one unavoidable perturbation and unconditionally departs from the base state. The determination of Re appeals to the socalled linear stability theory dealing with the evolution of fluctuations that are mathematically infinitesimal. It is indeed sufficient that there exist such a perturbation that is amplified for the base state to be unstable. It may happen that R, coincide with R,,in which case R 2 R, is a necessary and

10

Instabilities. Chaos and lhrbulence unconditional stability (global)

conditional stability

I

unconditional instability (local)

Fig. 1.2 Stability of the base state as a function of some control parameter R , e.g. the Reynolds number.

sufficient condition of instability but in general the interval [Rg,Rc]has a finite width, which defines a range of conditional stability: the stability of the base state depends on the shape and amplitude of the (finite amplitude) perturbations to which it is submitted. Most of the time this fundamentally nonlinear problem remains unsolved. In some favorable cases, one can succeed in computing the state that sets in above the threshold of the p r i m a r y instability by a perturbation method. This so-obtained state is then promoted as a new base state, the stability of which is of interest. This state can in turn become unstable with respect to a secondary instability that complicates the dynamics, and so on up to a chaotic state which, for fluid systems, is usually called turbulent. In this perspective, turbulence is considered as resulting from various modes at the end of a cascading process. Instead of staying at this formal viewpoint let us rather examine a particularly simple and intuitive concrete mechanism responsible for the instability of a liquid layer heated from below and initially at rest. 1.3.1

Thermal convection: the instability mechanism

Let us consider a layer of fluid at rest presenting a density stratification due to heating from below (Figure 1.3). A temperature difference AT = Tb - Tt > 0 is thus applied between the bottom ‘b’ and the top ‘t’ of the layer. Since, with few exceptions, the density decreases with increasing temperature and this stratification with heavier fluid on top of lighter fluid is potentially unstable in a vertical gravity field (like the ‘up’ position of the pendulum). When AT is small, the fluid stays at rest since the gravitational potential energy that would be gained by moving the heavier fluid to the bottom is not sufficient to counterbalance the energy loss by dissipation in that motion. More precisely, as long as the fluid is at rest, the heat transfer is entirely carried out by conduction. The temperature profile in the base


11

m

Fig. 1.3 Left: Instability mechanism of thermal convection. Right: Convection cells with wavelength A, = 2 n / k , .

state To(z)is obtained from the Fourier equation (1.14) that simply reduces = Tb-pz, where p = A T / h itself to d2To/dz2 = 0 since v E 0, hence To(z) is the applied temperature gradient, h being the height of the layer. Let us suppose a fluctuation localized inside a tiny droplet marked with a slightly higher temperature (perturbation 0 > 0). Since this bubble of hotter fluid is surrounded with colder denser fluid, it experiences an upward differential buoyancy force which makes it moving up, encountering ever colder fluid, which reinforce the motion: this is the destabilizing part of the mechanism. However, two stabilizing processes tend t o oppose this process. First, the so-induced velocity tends naturally to decay owing to viscous friction. Second, thermal diffusion aims at ironing out the horizontal temperature gradient accompanying fluctuation .'6 The fluid layer stays at rest as long as dissipation dominates but convection develops when the destabilization is sufficient, i.e. when AT is larger than some critical value AT, called the instability threshold. On the other hand, the stabilizing dissipative processes are diffusive in essence, so that their efficiency depends on the horizontal space dependence of the velocity and temperature modulations. Damping is fast on small scales and slow on large scales. It thus depends on the length-scale of the perturbation and it turns out that the onset of the instability corresponds to some optimum scale. For dimensional reasons this scale is of the order of the height h, as implied by the sketch in the left part of Figure 1.3. Accordingly the bifurcated convection state often sets in as a system of periodic rolls, with spatial period A, M 2h, as illustrated in its right part. The quantity Ic, = 27r/X, which appears when representing the plan-form in Fourier series is called the critical wavevector. The pattern formed has been called a dissipative structure by Prigogine [Glansdorff and Prigogine

(1971)l.

12

1.3.2

Instabilities, Chaos and !!hTbdence

Nonlinear convection and dynamical systems

Here we account for the dynamics of convection close t o the threshold by heuristic arguments that could be entirely supported by a rather complex detailed theoretical approach yielding the concrete value of the coefficients that we will introduce phenomenologically. The space-time coherence introduced in the system by the instability mechanism allows us to describe the evolution of the convecting layer using a simple variable A ( t ) measuring the amplitude of convection. We thus assume that, as far as their horizontal dependence is concerned, the velocity and temperature fields can be taken in the form

(vz,0) 0: A ( t )s i n ( k x ) ,

(1.15)

where A plays the role of an effective degree offreedom and k, is the critical wavevector introduced earlier. We are now interested in the phenomenological derivation of an evolution equation for this amplitude. At the infinitesimal stage, the perturbation is governed by a differential equation that should read

&A = uA,

(1.16)

where coefficient u accounts for the growth rate, negative below the threshold (damping) and positive beyond (amplification). Let us define a reduced control parameter T = (AT - ATc)/ATcand assume that the behavior of CJ as a function of r is not singular, so that its expression can be restricted to the first term of its Taylor expansion in powers of T : u = ?-/To,

(1.17)

where 70, homogeneous to a time, characterizes of the natural evolution of relevant fluctuations. Equation (1.16) where u is given by (1.17) accounts for the evolution of coupled velocity-temperature fluctuations as long as the amplitude A is sufficiently small. This does not raise difficulties when T < 0 since A decays but when T > 0, in the unstable domain, it is exponentially amplified and does not stay small for a long time. In order to model the nonlinear effects, we will assume that (1.16) remains valid but that u has t o be corrected and becomes a function of A itself. Noticing that the change ‘ A -A’ corresponds to a change of the spinning direction of the rolls (or to a translation of the structure by X,/2)2) and that the physics of the problem should not be sensitive to such a change, we are led to assume that the effective


13

growth rate (T,R is an even function of A. At lowest order of an expansion in powers of A, it can be taken as: Cetf

= (r - gA2)/T0 7

(1.18)

where g is called the Landau constant. In the simple case considered here, g is a positive quantity, i.e. g = 1/A2, A measuring the typical amplitude at which nonlinearities become effective. The experiments show (and the theory demonstrates) that the convection mechanism is indeed selflimiting, so that the effective growth rate decreases when the amplitude of convection increases. The nonlinear model accounting for convection in the neighborhood of the threshold then reads:

TO$A = F,(A) = T A- 9 A 3 .

(1.19)

Its study is elementary and will be resumed later in a more general context. Here it is our first example of effective dynamical system. It governs a simple scalar variable A representing the macroscopic evolution of our system. Upon rescaling t by T and A by A, equation (1.19) can be recast in a universal form: dt

A = rA

- A3 .

(1.20)

The evolution of A from an initial value A0 = 10-3 for r = 0.1 is illustrated in Figure 1.4. As long as r A >> A3 the growth remains exponential, which appears graphically linear on the lin-log plot in the left part of the figure. When A increases, the slope of the curve (T,R decreases. Ultimately, A saturates to a value A , = f i . Taking a negative initial value would have led t o -A,, owing to the symmetry of the equation. The graph giving the nontrivial state as a function of the control parameter, depicted in the right part of Figure 1.4, is called a bifurcation diagram. In the terminology of dynamical systems, variable A is the degree of f r e e d ~ m .Accordingly, ~ the real axis is here the phase space, while F,(A) is the vector field defined on this space that governs the dynamics. Stationary solutions A, are called fixed points, a stable fixed point being a simple example of attractor. The transition ‘conduction + convection’ is continuous with a stable nontrivial A, tending to zero as the threshold is approached from above. One then speaks of a supercritical bifurcation. Owing to the shape of the graph in Figure 1.4 (right), it is further called a fork bifurcation. Equation (1.20) is thus the n o r m a l f o r m for a supercritical fork bifurcation. 4Remember that we have left the field of analytical mechanics, see Note 2, p. 3

14


A

Fig. 1.4 Left: Evolution of the amplitude A as a function of time for model (1.20) with r = 0.1 (lin-log plot). Right: Bifurcation diagram giving the amplitude A of the steady state as a function of r . Conventionally, the unstable state A = 0 for T > 0 is indicated by dashes while a continuous thick line is used for the stable states, A = 0 for r < 0 and A = A . = *&for r > 0.

In the opposite case, i e . g < 0 in (1.19), the nontrivial solution appears below the threshold and is unstable, the bifurcation is then subcritical, and equation (1.20) has to be further completed to yield a meaningful model. This abstract approach will be resumed in Chapter 2 and 4. Convection rolls that develop beyond the threshold of the first instability can themselves become unstable with respect to other mechanisms. One thus has to study the flow that results from the superposition of the saturated primary mode to the initial base state, using a similar but technically much more complicated approach. Secondary modes can then be detected yielding, upon saturation, new base states ready for subsequent destabilization as the control parameter is further increased. The convecting layer is a continuous medium described by field variables. The reduction of the dynamics to a single scalar variable A , or a small set of similar variables when other instabilities have taken place, calls for justifications that will be examined in Chapter 4 as a precondition to the use of the concept of chaos for interpreting the increasing complexity of the dynamics all along the cascade of instabilities leading to turbulence. This approach will be all the more relevant when there are few convection cells, well confined by lateral walls at distances of the order of the height h itself. Its limitations will also have to be studied when confinement effects by far-apart lateral boundaries are too weak (Chapter 5).


1.3.3

15

Stability and instability of open flows

Natural convection has been the subject of many academic studies since it offers an ideal testing ground for ideas developed in nonlinear dynamics. However, systems of interest in engineering studies often display a supplementary feature: they involve open flows characterized by the existence of a mean current from upstream to downstream. Whereas the linear stability theory of open flows has a long history, dating back to the end of the Nineteenth Century, difficulties linked to this specific feature have delayed the nonlinear approach. Rather than a thorough account of results obtained in this field, Chapter 6 should thus be considered as a preliminary presentation aiming at better situating the problem in a nonlinear perspective and making its study easier. Purely kinetic effects play an important role in open flows and can by themselves already be at the origin of instabilities. Accordingly, a first distinction can be made between flows that are mechanically unstable and those that are mechanically stable. The former display an inflection point in the base velocity profile which makes them unstable in the absence of viscous dissipation according to Rayleigh’s criterion. Their prototype is the mixing layer illustrated in Figure 1.5 (left). By contrast, mechanically stable flows may become unstable only due to subtle feedbacks involving a transfer of momentum of viscous origin. A good example is given by the Blasius boundary layer flow depicted in Figure 1.5 (right). This classification will affect the whole process of transition to turbulence in an important way (Chapter 6, $6.2). A second distinction has to be made from the very existence of a global flow of matter entering the region of interest and leaving it at the outlet,

Fig. 1.5 Typical open flows. Left: Mixing layer created at the merging of two fluid veins with different velocities and maintained separated upstream by a splitting plate. Right: Boundary layer developing along a plate at some distance of its leading edge.

16

Instabilities, Chaos and lbrbulence

Fig. 1.6 Karmin vortex street emitted by a cylinder (courtesy P. Le Gal, IRPHE, Marseilles).

which makes implicit reference to a frame rigidly attached t o the laboratory (obstacle, rigid wall). An important difference indeed appears between instabilities said t o be convective, those that develop along the flow but are evacuated downstream and do not succeed in going upstream, and instabilities called absolute that are sufficiently intense to develop in spite of the global downstream transport. In the first case, the systems behaves as a noise amplifier and the result downstream essentially depends on the level of background fluctuations (residual turbulence). In the second case, one has rather to deal with a genuine self-sustained oscillator analogous t o those studied in Chapter 2. However, the situation is even more complicated by the fact that the intensity of instability mechanisms may vary in space. This is due to the fact that the velocity profile usually evolves downstream as a consequence of viscous dissipation that tends to smooth out the velocity gradients present at the entrance of the flow. Depending on the local intensity of the mechanism (linked t o the magnitude of the shear) one can observe a change of the character of the mode usually from absolute to convective, hence the possibility of bifurcated states localized in a given region of space and usually called global modes. The wake of a blunt body inserted in an otherwise uniform stream is such an example (Figure 1.6). A recirculation takes place close to the obstacle, rendering the overall downstream transport locally sufficiently weak that regularly shed vortices can develop and stay attached to it (KArmAn vortex street). The understanding of this combined problem ‘convective/absolute local/global + linear/nonlinear’ has made great progress recently but it requires sophisticated mathematical tools, especially in complex analysis, that would bring us far beyond the limited purpose of the present notes. The problem will just be evoked at a physically intuitive level in Chapter 6 .

+


17

In the above presentation the instabilities were tacitly assumed to be linear and supercritical. It may however happen that the bifurcation is subcritical and that several different flow regimes coexist in some range of control parameter, the bifurcated state saturating only “far from” the base state as for the plane channel flow. The base state may also happen to remain stable against infinitesimal fluctuations so that the nontrivial (turbulent) regime cannot be reached by perturbation. Examples are the pipe Poiseuille flow and the plane Couette flow. Though these flows are known to be linearly stable for all Reynolds numbers, they are expected to become turbulent as the shear rate increases beyond all limit. The transition indeed happens and take a rather explosive turn: turbulent bursts developing intermittently from localized finite amplitude perturbations show up during the process, coexist with still laminar flow and then merge to fill the systems with fully developed turbulence. Figure. 1.7 displays such a turbulent spot in plane Couette flow. It shows that the pocket of turbulence is not structureless but, on the contrary, present a streamwise streaky pattern. This type of structure turns out to be omnipresent in turbulent wall flows. Their production and sustainment mechanisms are still the subject of current research (Chapter 6, 56.3.4).

Fig. 1.7 Mature turbulent spot induced by a localized perturbation of finite amplitude in plane Couette flow (courtesy S. Bottin and 0 . Dauchot, GIT, CE Saclay).

1.3.4

Beyond the transition: filly developed turbulence

Problems considered up to now all relate to the steps of the cascade leading from a simple and regular base state to a complex and irregular flow still partly ordered but chaotic, that may not yet be called turbulent. It seems indeed important to understand the steps of the laminar-turbulent transition just sketched in a deterministic framework. The underlying aim is of course to control it as best as we can, delay it if it is harmful, or advance it

18


if we need better mixing. However it turns out that beyond the transition we have t o change our mind and take a fully statistical viewpoint. As a matter of fact, when we pull the considered system (flow) ever farther from equilibrium, more and more degrees of freedom become excited and it no longer makes sense to focus on their individual dynamics, like when passing from few-body systems well described by analytical mechanics to gases for which the thermodynamic approach is more appropriate. This remark could lead to understand developed turbulence as a new macroscopic state of matter at a scale where the fluid would have transport properties very different from those of ordinary fluids in their laminar state where molecular chaos still control diffusion. Things are unfortunately less simple. The usual distinction between microscopic and macroscopic scales works well for thermodynamics because there is a wide gap between them, the precise reason for which the concept of material point (or fluid particle) makes sense. By contrast, in turbulence, relevant scales belong to a continuous range from the size of the flow domain to small scales. While the smallest eddies seem to evolve randomly like molecules in a gas, they are still coupled and, in fact, driven by the scales above them so that no decoupling is truly legitimate. In Chapter 7, we will approach the theory of turbulence in a very preliminary way only, without trying to compete with numerous excellent books dealing with it, from conceptual problems to applications. Our limited aim will be to illustrate the idea of an inertial cascade (cf. Fig. 1.8) transferring energy from large scales where instability mechanisms generate the eddies, down to the smallest scales where viscosity successfully irons out the fluctuations. It is not difficult to catch an idea about this transfer by the advection term v . Vv of the Navier-Stokes equations. Considering an sin(lcz), eddy motion locally described by a simple trigonometric line 21, we get sin(lcz)a, sin(lcz) = ksin(kz) cos(lcz) = iksin(2kz) that presents N

production

transfer

dissipathn

Fig. 1.8 Illustration of the Kolmogorov cascade from large scales to small scales where fluctuating eddies are blurred by viscous dissipation.


19

itself as a source term in the evolution equation for a mode with a spatial scale half of the initial one. The result of this transfer is a repartition of energy according to the celebrated k P 5 l 3 Kolmogorov law that will be derived heuristically. In a second part, we will turn t o the concrete problem of predicting the lowest order statistics of specific turbulent flows, obtaining the so-called Reynolds averaged equations governing the mean flow, which immediately opens the problem of the closure of the statistical description. As a matter of fact, the averaging of primitive equations introduces new higher order statistical quantities called the Reynolds stresses that remains to be evaluated in one way or another. We will then exploit the already mentioned analogy between the kinetic theory of gases and the statistical theory of turbulence to introduce a disputable but heuristically valuable approach t o turbulent flows resting on the concept of eddy diffusivity patterned on that of molecular diffusivity (see Figure 1.9). This will serve us t o make a first evaluation of the average properties of a turbulent flow, taking the turbulent boundary layer as an example, and obtaining the classical KBrmBn logarithmic law. The main shortcoming of this approach derives from the absence of scale decoupling mentioned above but we will not do much more than mentioning it, suggesting further that progress can be obtained in concrete situations by numerical simulations, and observing that the latter always imply the crucial step of sub-grid-scale modeling, i. e. the modeling of smallest scales that cannot be explicitly accounted for in the numerics.

Fig. 1.9 Visualization of a turbulent flow downstream a grid. The fluid passing through a given mesh is seeded with a tracer that demonstrates turbulent diffusion in the flow. After a color picture by J.L. Balint, M. Ayrault and J.P. Schon (Ecole Centrale de Lyon), courtesy M. Lesieur, in “La turbulence d6veloppCe,” La Recherche, no 139 (1982), with permission.


20

1.4

Conclusion

Numerous books develop the topics to be touched upon. A partial list is given in the bibliography. At the risk of repetition, let us emphasize that we have tried to present a first approach to current problems, using a wide range of techniques, linear, nonlinear, deterministic, statistical, each with its own qualities and limitations, in order to lucidly face complicated situations encountered in our familiar environment, industrial or natural. Accordingly, in the concluding chapter we will evoke the problem of the Earth’s climate as a concentration of the kind of topics examined at one or another moment, with special reference to the predictability problem and modeling issues involved in this nonlinear dynamical system with heterogenous space-time scales. Appendix A recalls some elementary and not so elementary results of linear algebra, while Appendix B works in roughly the same direction by giving rudiments of numerical simulation techniques expected to be of help in the understanding of complex processes (provided that we are able to build simplified models with valuable metaphoric value).

1.5 1.5.1

Exercises

Chemical reactions

Chemical kinetics is a field that has contributed much to the development of applied nonlinear dynamics in out-of-equilibrium systems. Here we consider a simplistic kinetic model introduced by Prigogine and Lefever in 1968, called the Bruxellator and accounting for a hypothetical reaction between two components A and B with two end products C and D, four steps and two intermediate compounds X and Y. Concentrations A and B of reactants A and B are the control parameters whereas those X and Y of the intermediate species X and Y are the variables. The global reaction: A+B + C+D can be decomposed into successive steps: A + X, B + X + Y+C, 2Xs-Y + 3x, X + D. Assuming that the kinetic constants of each reaction is equal to 1, write down the equations for each step and show that the reaction is governed by X =A -(B

+ l)X +X’Y,

Y =BX -X2Y

This system will be further studied in Exercise 4.6.7.


21

Elements of chemical kinetics. Consider: XniAi

-+

Xn:Ai

an elementary step between reactants Ai with concentration Ai [if specie Ai is absent from the left (right) hand side then ni = 0 (n: = O ) ] . The reaction rate corresponding to this step measures the number of reacting collisions per unit time. A collision is the meeting at a single physical point of all the components on the 1.h.s. producing all those of the r.h.s. in the proportions given by the stoichiometric coefficients. The probability for reactant Ai to be there is proportional to the concentration and, if one neglects the correlations between species, one expects the probability of the collision as a product of individual probabilities and thus a reaction rate k A?, involving only participants on the 1.h.s. and where k is the corresponding rate constant. When such a reaction takes place, the number of molecules of component Ai varies by n: - ni, so that one gets:

ni

A:' .

Ai = (n: - ni)k i

The total variation of the concentration of a given specie Ai for a compound reaction is the sum of the variations at each step. An oscillatory reaction, discovered by Belousov in the fifties and further studied by Zhabotinsky in the sixties (BZ reaction) has been the subject of intense laboratory work. It corresponds to the oxydization of a organic reducer (malonic acid) by BrO, ions with a redox couple ( e . 9 . Ce3+/Ce4+) as a catalyst. It involves about 15 chemical species coupled by an equivalent number of intermediate reaction steps. Simplified models of this reaction have been devised with only 5 steps and 3 intermediate variables (Field et Noyes, 1974). 1.5.2

Prey-predator systems

Prey and predators interact in much the same way as the molecules considered in the previous exercise. The dynamics of the population X of some specie X is usually described in terms of an effective growth rate:

x

= (Y&X

(1.21)

where the expression of (Y,E accounts for the balance between birth and death, which depends on control parameters and the population itself.

22

Instabilities, Chaos and n r b d e n c e

1) Assume that the effective growth rate is simply proportional to the amount of available food and interpret the different terms appearing in a ,= ~ a - P X , where a and ,B are two positive constants. The corresponding evolution equation is usually called the (continuous-time) logistic eqvation. What is the meaning of quantity X , = a / P . Derive the solution X ( t ) of this ‘limited growth’ model starting with initial condition X = X O > 0 at t = 0 by explicit time integration for X O < X , and X O > X,. Extract the long-term behavior ( t 4 co). Observe also that the evolution of X here is the same as that of A2 where A is governed by (1.19) for r > 0, apart from scale changes on X and t to be determined [see Figure 1.4 (left) that would correspond to the case X O < X,]. 2) In the model considered above, X is the population of a predator X and the available food is determined by the population Y of its prey Y , the food is thus not indefinitely available and the term -,BX is no longer necessary. A simple assumption is that the growth of the population is associated with a food consumption that is proportional to the presence of prey (the equivalent of a ‘reactive collision’ in chemistry) and that its decay is due t o some natural death rate. Argue in favor of expression a& = CU‘Y - Lug

for a , in ~ (1.21). What should be the signs of the constants a’ and ao. 3) In order to close the system we need to write down an equation for the effective growth rate of Y . By arguments similar to previous ones, justify the expression Y =yY -6 X Y . This coupled system will be studied in Exercise 2.5.12. Try t o generalize the approach to describe more complicated prey-predator models, called Loth-Volterra systems, by changing the assumptions, e.g. that the growth of the prey is also limited by competition for food.

1.5.3

Digusion equation

We are interested in the decay of a solution to the diffusion equation [e.g. the heat equation (l.S)] in space dimension one: (1.22) starting from some initial condition &(x).


23

1) Determine by substitution the relation between the growth rate s and the wavevector k of a periodic fluctuation taken in the form 8 ~ ( xt,) = 6exp(st) exp(ilcz) and derive its evolution as time goes on. 2) Adding boundary conditions 8 ( 0 ,t ) = 8 ( l ,t ) = 0, solve the evolution problem formally for a solution starting with an initial condition that can 8,,sin(rnz/C). Extract its be expanded as a sine series: d ( z , 0) = Cr=p=, asymptotic solution in the limit t + 00. 3) Going back to the infinite medium, consider functions 8 that are taken in the form 8(z, t ) = exp(aw(z,t ) ).

(1.23)

Derive the partial differential equation governing w(z, t ) , and next that governing v = &w. Find the value of a bringing this equation in the form

a,v

+ vdxv = Icdxxv

(1.24)

called the Burgers e q ~ a t i o n The . ~ inverse of function change (1.23), called the Hopf-Cole transformation, offers an exact analytical way to linearize (1.24) and to put it back in the form (1.22).

~~

~

5J.M. Burgers, The nonlinear diffusion equation (Reidel, 1974), see U. Frisch & J. Bec, “Burgulence,” in [Lesieur et al. (2001)l for a recent review.


Chapter 2

First Steps in Nonlinear Dynamics

In this chapter we consider the dynamics of systems with a very small number of variables, as a prerequisite to the study of chaos, postponed to Chapter 4 after an examination of convection in Chapter 3, which will give us a physical motivation.

2.1

2.1.1

From Oscillators to Dynamical Systems

First definitions

Newtonian mechanics is the archetype of deterministic dynamical theories. Governed by an equation of the form

where X represents the position of a particle with mass m submitted to a force F , it accounts for processes that are invariants under a change of the arrow of time. A traditional example of linear system is the harmonic oscallator (Fig. 1.1, left, p. 3) that describes the motion of a mass attached to an ideal spring with a restoring force proportional to the extension F = - k X (Hooke’s law). This elastic force, of internal origin, derives from a potential:

depicted in Figure 2.1 (left). Here we have

mX

+kX 25

= 0.


26

Fig. 2.1 Left: Potential for the harmonic oscillator. Right: Vector field and trajectories in the phase plane, the plane of variables Xi E X and X z P.

The evolution is uniquely determined when initial conditions are specified:

X = X(O)

X =V(O)

and

at

t =0.

(In the following, we will systematically use the superscript ‘(0)’t o denote initial conditions.) The solution then reads:

Right now, it turns out advantageous to substitute a geometrical description (Fig. 2.1, right) to this analytical description. So, let us consider trajectories in the phase space where coordinates are the position X and the momentum

P = mV = m X .

(2.3)

Equation (2.2) now reads

One can trace back the origin of this representation from the need to turn a high order differential equation into a first order differential system. On general grounds, this is done by introducing the successive derivatives as intermediate variables. For example, equation: dnX/dtn = F(dn-lX/dtn-lX,. . . ,X , X , X ; t ) is reduced to a first order system by setting X1 ;= X, Xz X, 3 dn-lX/dtn-l, which yields: x1

= x2,

xa

= x 3 , . ..

,

(2.5) X , . . .,

X n = F ( X n , . . . ,x3,xz,x1; t).

2. First Steps in Nonlinear Dynamics

27

This operation is preliminary to any numerical implementation in view of simulation (Appendix B). There is a slight disadvantage to start from mechanics’ for an introduction of basic concepts of the theory of dynamical systems since Newton’s equations endow the phase space with a specific structure and give it an even number of dimensions, d = 2 n . As seen above, this space is indeed constructed as the product of the configuration space (variables X i , i = 1,.. . ,n ) and of the m o m e n t u m space (variables Pi,i = 1,.. . ,n),each pair of conjugate variables ( X i , P i ) forming a degree of freedom. In the general case when the state of a system is specified using a supposedly sufficient number of variables, no longer grouped by pairs, we will call any of these state variable a ‘degree of freedom’. Accordingly, the ‘phase space’ will then be the space where these variables “live” and the dimension of this space will just be the number of these variables. In agreement with the intuitive concept of d e t e r m i n i s m , the dimension d is also exactly the number of conditions necessary to specify any evolution uniquely. A way t o avoid ambiguities would be to keep the terms ‘phase space’ and ‘degrees of freedom’ t o mechanics in a strict sense and to use state variables and state space in all other cases, which we will not do since the risk of confusion is limited. Anyway, let us consider the general case of a phase space X with dimension d spanned by d variables { X I , . . . ,X d } z X. The evolution of the so-defined variables is governed by a system that symbolically reads

x = 3 ( X ; t ),

(2.6)

where 3 is a set of d functions representing the components of a vector field defined on X that specifies the “velocity” of the point representing the system in its phase space. When the properties of 3guarantee the existence and uniqueness of the solution to the initial value problem, in practice when the vector field 3 is C1 (differentiable with continuous first derivative), one says that it defines a flow on phase space. When t is explicitly absent from the definition of 3,the system is said to be a u t o n o m o u s , otherwise it is forced. In practice, among forced systems, only periodically forced systems will be of interest to us, i.e. systems such that 3 ( X ; t + T ) s F ( X ; t ) for some minimal time interval T called the period. Within this class one often distinguishes parametric forcing for lSee later, 52.1.2, for a brief presentation of the formalism of analytical mechanics.

Instabilities. Chaos and Turbulence

28

which the expression of 3 changes in time, e.g. the parametric linear oscillator (Mathieu equation): X

+ (1+ a sin(wt>>x= 0,

T = 27r/w,

from external forcing where an otherwise autonomous system is submitted to a periodic force independent of its state, i.e. X

+ x = fsin(wt).

Noisy systems can be understood as particular forced systems with a random forcing. In case of additive noise, this defines the so-called Langevin equation

x = F(X)+ E ( t ) , where E ( t ) is a random vector function. Quite different tools of statistical essence are then required which will not be introduced here since we want to stick to the deterministic point of view. During its evolution, the system follows a phase space trajectory starting at X(O) when t = and obtained by integration of (2.6):

X(t) = X(O)

+

l(o) t

F(X(t');t')dt'

The orbit is the set of points in X visited by the system in the course of a given trajectory. The description of a system's dynamics in terms of sets of orbits, called its phase portrait, as a function of its control parameters, is the field of qualitative dynamics. Relation (2.7) allows us to define a map of X onto itself. Upon specifying an integration time T , we get a time-T m a p

X(t

+ T ) 5 +r (X(t)) = X(t) +

1

t+r

dt' F(X(t');t'),

and, starting with an initial condition X(O),we obtain a discrete sampling = of the trajectory, XO x('),X I = ' P ~ ( X O~2) , = + r ( ~ l ) ,... , +r(Xk)*.. The time-T map is really interesting only when T corresponds to some characteristics of the system. The most important case corresponds to periodic forcing with period T = T , in which case performs a stroboscopic analysis of the dynamics, i.e. takes pictures of the system at the period of a strobe signal in phase with the forcing. Later we will see another related way to arrive at such discrete-time systems as already mentioned in


29

Chapter 1, p. 4. The concepts of trajectories and phase portraits transpose immediately t o systems written as first-order iterations Xk+l = +(Xk; k). From a practical viewpoint the determination of the trajectory issued from some initial condition X(O) by numerical integration can be viewed as resulting from the iteration of a map +T integrating the field 3 over a time interval r = At chosen from accuracy considerations specific to the numerical scheme (cf. Appendix B). For the moment, let us consider autonomous systems and especially the simple case of the harmonic oscillator. Its phase space is the plane ( X ,P) isomorphic to R2. The right part of Figure 2.1 displays the corresponding vector field.2 In this representation, trajectories follow elliptic orbits. On general grounds it turns out useful to scale the variables as much as possible in order to cast the system into its most universal form, hiding its specificities inside the details of the variable change. Here such a transformation simply yields x 1

= x2,

x 2

= -XI.

(2.8)

Reversibility is one of the fundamental characteristics of ideal mechanical systems (ie. without friction). The change ‘t I+ -t’ indeed leaves the Newton equations invariant. This property is associated to energy conservation. Defining E = $ X ; $ X i and computing d E / d t using (2.8) one can check that E is a constant of motion. Things are different for a damped system that dissipates its energy. At a linear stage, the introduction of a viscous friction proportional to the rate of change of the variable leads to

+

x +2 7 x +x = 0 ,

(2.9)

where 7 > 0 measures the strength of the damping. Here the change t C ) t/wo has been performed in order t o normalize the period of the ideal oscillator to 27r. Trajectories are obtained in parametric form as

X ( t ) = Xexp(-vt) cos(wt - cp), where X and cp can be computed from the initial conditions X(O) and at t = 0 by identification, and where w = (1 - q2)l/’ corresponds to the angular frequency of a damped oscillation only when the friction is 2Here it has been obtained using the MATLABmacro quiver, explicitly: X=[-O. 8 : O . 1 :O. 81 ; U=ones( 1 , size(X.2)) ; P=X’ ; FX=P*U; FP=-U’*X; quiver (X,P,FX,FP).

30


e-

-0.8 -0.8

Fig. 2.2 Vector field of the non-ideal linear oscillator in its phase plane ( X I ,X z ) for 17 = 0.5. Left: Damped oscillator (17 > 0). Right: Driven oscillator (17 < 0).

sufficiently weak, i.e. when 77 < 1 (the over-damped case 77 > 1 will be considered as an exercise). In the phase plane, equation (2.9) reads XI = xz ,

x 2

= -277x2 - XI,

(2.10)

and the orbits now take the aspect of spirals converging towards the origin, see Figure 2.2 (left) that displays the corresponding vector field. The instantaneous dissipation rate of the total energy, still defined as E = $ ( X l + X i ) , is now given by dE/dt = -2qXi < 0. For an excited system that would receive energy from the exterior world by viscous driving, thus governed by (2.9) but with 77 < 0, the integration of the vector field would give diverging spiral trajectories easily imagined from the vector field depicted in Figure 2.2 (right). In fact, when defining a state physically, one never considers a mathematical point in phase space but rather a small, physically infinitesimal domain around this point. This remark suggests we focus on the future of sets of systems that at a given time belong to volume elements in phase space (surface elements in the specific two-dimensional case of interest here). For a general dynamical system (2.6) defined on a d-dimensional phase space X,one then shows that the local evolution rate of the volume A of an infinitesimal domain around some point X is given by the divergence of the vector field 3 computed at this point (cf. Exercise 2.5.1): (2.11)

For an ideal oscillator (2.8), one immediately obtains that the divergence is


31

Fig. 2.3 Evolution of volumes in phase space. Left: Conservative system. Right: Dissipative system.

zero: the surface of a small surface element is thus conserved (Fig. 2.3, left). This property can be generalized to all frictionless mechanical systems, that are called conservative for that reason. The same calculation for the damped oscillator leads to div 3= -2q < 0, thus to an indefinite erosion of areas (Fig. 2.3, right). This reduction is characteristic of dissipative systems whose permanent regimes, asymptotic states at the limit t + 00 after the damping out of transients, are described by attractors. As suggested above, the phase space volumes give a measure of the number of accessible states. Asymptotically the damped oscillator always ends at its rest position X I = X2 G 0, a single state whatever the initial energy. The attractor is here a fixed point in phase space, ‘fixed point’ because the orbit of the trajectory starting there is just reduced to that point. By anticipation, we can say that it is a stable fixed point since trajectories starting in its vicinity converge to it as t tends to infinity. By contrast, the fixed point at the origin of the phase space associated to an driven oscillator in not an attractor but a repellor, it is unstable and trajectories move apart from it. General systems are not conservative. They can be fueled in energy in some regions of their phase space while dissipating it in other regions. This feature, that cannot be achieved in the framework of linear systems with constant coefficients (sign of d i v 7 fixed once for all), will turn out to be essential to the existence of self-sustained oscillations and more complicated time behavior.

32

2.1.2

Instabilities, Chaos and Thrbulence

Formalism of analytical mechanics

In order to ease the solution to some exercises, we give here an introduction to the analytical formalism of classical mechanics that allows one to pass from the Newton equations (second order in time) to the Hamilton equations (first order in time) giving to the intermediate variables so-introduced their status of conjugate momenta to the generalized coordinates. For a system of n N (subscript ‘N’ for Newton) material points with masses mi and positions Xi E (xi,yi, z i ) submitted to forces fi deriving from a potential V({Xi}; t ) ,the Newton equations read: ,nN).

(2.12)

The kinetic energy is defined by: 1 nN 2 z=. 1

7= -Cmi(xz)2, and one easily checks that the total energy

is conserved. But this so-called Newtonian formulation makes the treatment of systems with constraints somehow awkward. The Lagrangian formalism answers this problem by introducing n~ (‘L’ for Lagrange) generalized coordinate^.^ Let

be the change of variables from the Xis to the q j s expressing the constraints. The Lagrangian is then defined by

L=L(q,q)= 7 - V , so that (2.12) can be rewritten as a set of Lagrange equations: (2.14) 3 n 2 0 for any ( X , Y )# (O,O)).I ! (distance t o the origin) start2b) Determine the evolution of 2 = &? ing from some initial condition Zo > 0 at t = 0 and show that that Z diverges at t, = 1 / 2 0 , i.e. the corresponding trajectory spirals away t o infinity in a finite time. 3) We now search t o extend the concept of energy used up t o now, in such a way that the study of its variations allows one to decide about stability or instability of a given state, conveniently taken as the origin of coordinates:

+

Definition: Let G(X) be a function of point X in phase space X,taking its values in R’, definite positive, i.e. such that G(0) = 0 and G(X # 0) > 0 (in practice a definite positive quadratic form). A sufficient condition of asymptotic stability, is that the amplitude of the perturbation, as measured by G decreases in the course of time and tends to zero as t goes to infinity. This will be the case if 4 = C j &,G X j = C jax,G 3 j is negative definite as a function of X E X taking its values in R, such that o(0) = 0 and G(X # 0) < 0. Such a G is called a Lyapanov f ~ n c t i o n . ~ 0

’This definition can be extended to the case of continuous media described in terms of fields (velocity, temperature, concentration,. . . ), in which case the Lyapunov function becomes a functional of the fields.


63

Application: Consider the system defined as:

X=XY-X3, Y = -Y - 2 x 2 . Show that the origin is the unique fixed point and further that it is marginally stable. Since no conclusion can be drawn at the linear stage, by generalizing the energy E to a function in the form:

q x ,Y ) = ;(ax2+ BY2), determine the conditions under which E is a Lyapunov function, thus proving the asymptotic stability of this state. 2.5.3

Linear stability and bifurcations in dimension three

As a useful preliminary to Exercise 3.3.3 in Chapter 3, consider a real, three-dimensional, linear dynamical system in the form 3

j=l

1) Recall the relations between the coefficients and the roots of its characteristic equation written as s3 us2 bs c = 0. Further identify all the possible non-degenerate or degenerate cases (real or complex nature of the roots, their sign or the sign of their real part, simple and multiple roots). Write down the Jordan normal form of the operator corresponding to each situation (see Appendix A, 5A.2 for a reminder) [Answer: see Fig. A.l(b)].

+

+ +

2) When the control parameters vary, the eigenvalues of the system move in the complex plane. Determine the remarkable relations fulfilled by the coefficients of the characteristic equation in the marginal cases and sketch the spectrum of the operator in the complex plane before, at, and after a stationary bifurcation (s = 0) or an oscillatory bifurcation (%(s) = 0). [Answer: see Fig. 2.14.1 2.5.4

Coupled linear oscillators

Consider the following system of two coupled oscillators

X + X = Y ,

Y+R2Y= -cx.

Instabilities, Chaos and Tvrbulence

64

'

stationary bifurcation stable

oscillatory bifurcation

0

real spectrum

I

I I

1

complex spectrum

Fig. 2.14 Typical bifurcations in the three-dimensional case, Exercise 2.5.3.

Find the equation governing the angular frequencies of the eigenmodes taken as ( X ,Y )= ( X o ,Yo)exp(iwt), observe that the solutions appear as pairs of opposed eigenvalues. Determine the domain in the plane of parameters R2 and c where the eigenmodes are stable. [Hint: remember that for a mechanical system, stability means orbital stability, with purely imaginary eigenvalues.] 2.5.5

Logistic equation

Reexamine the limited growth model (logistic equation) introduced in Chapter 1, Exercise 1.5.2 about population dynamics. Study the stability of its two fixed points and draw its phase portrait. 2.5.6

Dynamical systems and solitons

1) Consider a dynamical system with a single degree of freedom in the sense of analytical dynamics (a pair of conjugate variables) with a force deriving from the potential x 2

V ( X )= -2

x3 +3

(2.69)

65


I

1.5,

..

-’40

-0.5

0:o

0.5

l:o

i.5

’

Fig. 2.15 Left: Phase portrait of the system with a force deriving from potential (2.69) Right: Profile of the soliton corresponding to the separatrix on the left part.

la) Write down the second order differential equation governing X (mass m = 1) and then the first order system for X and P = X . Find the fixed

points, compute their eigenvalues and eigenvectors (if any). Sketch the phase portrait in the phase plane ( X ,P ) [Answer: Figure 2.15, left]. l b ) Draw the graph of the potential V ( X ) and determine the energy corresponding to the separatrix, the special trajectory that limits the domain of orbits bounded around the minimum of the potential. Find the turning point and the equation accounting for this trajectory. Determine its solution by identification with X(t) = Xo/ Cosh2(t/7). Why is this form “natural”? lc) Sketch the phase portrait of the system when perturbed by the introduction of a weak viscous friction, taking the weakly damped pendulum as an example. 2) The Korteweg-de Vries equation reads:

This is a non-dissipative equation with a nonlinearity of hydrodynamic type, identical to that of the Burgers equation considered in Chapter 1, Exercise 1.5.3. It is integrable and its solution can be expressed in terms of a superposition of interacting solitons. Here we are interested in a solution moving without with a single soliton such that h + 0 when x + &XJ deformation at speed c, and thus only function of the combination = x-ct.

0 implies &v, > 0, i.e. an upward acceleration as guessed intuitively.

Heat equation The Fourier equation must be written for a fluid particle (cf. p. 7) since, as already mentioned, it is its advection in a spatially varying temperature field that plays the essential role in the feedback loop, hence: = &T + v * VT = d Z 2 T . ZT d Expanding the term v . VT to first order (linearization) we find:

w%[To(.z)+ 81 = v*azTo(z)= -vzP

,

which comes from the temperature field at order zero. This leads to:

ate = ndx28 + pv, .

(3.4)

Remarks Lateral boundary conditions have not yet been specified. Here, we tacitly assume that we deal with a horizontally unbounded layer, or at least that the horizontal dimensions are large when compared to the sole characteristic length in the problem: the height h of the layer. Moreover, the horizontal velocity component and the pressure are absent from the problem at this stage. In fact they are only indirectly coupled to vz and 8 by the need to insure the continuity of the fluid and the closing of flow lines. The model is thus highly simplified. This deficiency will impede us to determine the critical wave-length that will thus be fixed by dimensional considerations. Anyway, we shall now illustrate the extension to continuous media of the stability analysis introduced at the beginning of the previous chapter using the simplified model (3.3, 3.4).

75

3. Life and Death of Dissipative Strvctures

3.1.3

Normal mode analysis, general perspective

The system formed by Eqs. (3.3, 3.4) is typical of linear stability problems in continuous media, i.e. a system of linear partial differential equations (with constant coefficients in the simplest case). It presents itself as an initial value problem for the perturbations that we write formally as:

atv = L,(a,,

.. . ) V ,

(3.5)

where V represents the set of perturbations. The linear operator L,, here solved for at and first order in time, contains spatial partial derivatives (az,.. . ) and also depends on a set of control parameters denoted as r. In general, the instability can be controlled using a single quantity that can be varied from the outside (for convection, it is simply the applied temperature gradient p), all other parameters of the system being kept fixed. Problem (3.5) is linear. Its solution can therefore be searched by means of a superposition:

V(X,t ) =

c

AnXn(x,t)

(3.6)

n

that is further introduced in the differential problem. Setting:

X(X,t) = e x p ( s t ) k ( x ) ,

(3.7)

and inserting this assumption in (3.5), we get sX(x) = L,X(x) .

(3.8)

The stability study then comes to an eigenvalue problem. The states X ( x ) are called the normal modes of the problem. These modes have spatial structures that are the mathematical expression of the physical coherence of the processes at work in the system. The nature of the spectrum of L,. depends on the applied boundary conditions (Exercise 3.3.1). When the system is unbounded in some directions of space, the spectrum is formed with continuous branches, indexed by as many continuous “separation” parameters as unbounded directions. For example if the mechanism singles out a specific direction (‘vertical’ in the case of convection), and if the system is invariant under translations in two complementary directions (‘horizontal’), performing a Fourier transform, one looks for normal modes in the form: Xn(Z,y,

+

e x ~ ( i ( k z ~kyy))%(z)

1

(3.9)


76

which defines the wavevector k l = (kz,ky) as a separation parameter. Under the substitution VI = (ax, 8,) I-) i k l , the operator C, = C,(a,, . . . ) is transformed into an ordinary differential operator in z , the only remaining independent variable, i.e. C T ( i k xik,, , d/dz). The wavevector k l thus serves t o label the eigenmode branches in addition t o a discrete index related to the confinement direction z. Equation (3.8) has nontrivial solutions provided that the growth-rate and the wavevector k l , introduced in (3.7) and (3.9) fulfill a compatibility condition = sn

(T, k

l)

(3.10)

called the dispersion relation. Before developing this approach on the simplified model of convection in Section 3.1.4, let us note a few points relative t o confinement effects to be re-examined at the beginning of Chapter 4. When the system is rotationally invariant in the plane orthogonal t o the direction in which the mechanism is operating, then s, can only depend on k = lkll and not on its orientation. When the system is bounded in two space directions, e.g. y and z , translationally invariant in the last one x, the continuous component k, is replaced by a discrete index. When the system is bounded in all three directions, i.e. when its size is of the order of the scale over which the instability mechanism is operating in all directions, the spectrum loses its last continuous dependence on k and becomes fully discrete. The eigenvalues are all distinct, except for degeneracies linked to physical symmetries.’ A situation close t o that considered in the previous chapter is recovered but with infinite series of eigenvalues and normal modes to acknowledge the fact that we deal with continuous media with infinitely many degrees of freedom. Assumption (3.6) brings us back to the formulation initially introduced at the end of Section 2.2, p. 35. Considering the evolution of the solution corresponding to a “pure” mode with index n and amplitude A , (supposed to be infinitesimal) we have

V

= A n X n ( x ,t) = A:) exp(s,t)X,(x)

,

where A:) is the initial condition for the amplitude of mode n. Let us recall that the eigenvalue is a priori complex since the spectrum is entirely ‘In this case the spatial structure of the normal modes is specific to the geometry considered and mirrors spatial resonance properties of the mechanism with the shape of the set-up.


77

real only when the operator can be made self-adjoint for some well-chosen scalar product (see Appendix A). Let us consider the case of an unbounded system with a single continuous separation parameter k and write s n ( k , r )= U , ( ~ , T ) - i w , ( k , r ) . Given T and k, we can order the modes by decreasing values of their real part un(k,r ) , which allows us from definition (3.7) to distinguish between stable normal modes with u, < 0 (damped) and unstable modes with B , > 0 (amplified). The dissipative character of the medium implies that modes with the shortest wave-lengths are strongly damped, i.e. un(k) + -00 when k + 00. The system bifurcates against a given mode n with wavevector k when, upon variation of the control parameter T , this mode goes from stable to unstable. The marginal conditions for this modes, superscript ‘(m)’, are defined by the condition u n ( k , r )= 0, that makes it neutral, which can be written as: T

=T

p ( k )

when solved for T . Let us suppose that, as in convection, increasing the stress corresponds to increasing the control parameter T , the marginal curve for some mode n usually reaches its minimum for some k = k p ) called the critical wavevector for that mode. The corresponding value of T , rn - T (m) n (kt)) is the corresponding threshold. Now, according to the general discussion about stability in the previous chapter, linear instability takes place as soon as one normal mode becomes unstable. Accordingly, the linear instability threshold rC is the minimum over n of all the so-defined r k ) , achieved for, say, n = n,. The wavevector of that mode called the critical wavevector of the instability, hence characIc, = kk). terized by the set n,, T , = Apart from its growth properties, the rest of the time dependence of a mode depends on the value of w , ( k , r ) . When w, = 0, the mode is said to be stationary, whereas when w, # 0, one speaks of an oscillatory mode. The value of the angular frequency at threshold, wc = wnc (kc,r C ) , thus allows one to distinguish stationary from oscillatory instabilities. The classification of instabilities according to the spatio-temporal structure of their critical mode will be reexamined in 53.1.6.

~g),

78

Instabilities, Chaos and lhrbulence

3.1.4

Back to the model

Let us come back to the simplified model (3.3, 3.4). We assume that the fluid layer is unbounded in the x direction so that, according t o (3.7-3.9), solutions are searched in the form {vz,0) = {V,0 ) exp(st) exp(ikx). We obtain:

s v = -vk2V

so = - K k 2 0

+a g o , + pv,

which is in fact a homogeneous algebraic system of two equations for two unknowns: (s

+ vk2)V - a g o = 0 ,

-pv+(s+Kk2)o

=o.

The system has non-trivial solutions only if its determinant cancels: (S

+ v k 2 ) ( s+ r;k2)- a g p = s2 + (vk2+ d 2 ) s + ~

v -kagp~ = 0 . (3.11)

This compatibility condition linking the growth-rate s and the wavevector k of the perturbation is here the expression taken by the dispersion relation (3.10). We get a single branch (and thus no discrete index as alluded to above) since the differential problem in t has been replaced by an algebraic system, due to our neglect of the z-dependence of the fluctuations. If the real part of s(k) is negative, the mode is damped and the layer is stable against a perturbation with wavelength A = 21r/k. Otherwise the fluctuation is amplified and the mode k is unstable. Let us try to determine the threshold from (3.11). Here it is a quadratic equation in s that can have two real or complex roots.2 The discriminant

A = (v k 2 + K L ~ )+ 4 [ a g p - ( v k 2 )( rck2)] = (v k 2 - &k2)

+ 4agP

can be negative, and the corresponding solutions of (3.11) have non-zero imaginary parts, only when p is sufficiently large and negative, ie. according to our conventions, in case of strong heating from above, but in this case the modes are always damped since the sum of the roots

s = -;(vk2 + K k 2 ) 'Let

5-1,'

be the two roots, one has 0 = (s - s1)(s

- 52)

(3.12) = s2

-

(sl + 5')s

+ slsZ =

s 2 - Ss

+ P where S is the sum of the roots and P their product. The discriminant is

A = S'

- 4P

and the roots are s1,2 = s* = i(S f 6).


79

is then negative. This analysis thus confirms the intuition according t o which heating must be from below in order to have an instability. Moreover, if there is an instability, it can only be stationary, with two real roots, one positive, the other negative since their sum is negative. In order to determine the sign of the roots we have just t o consider their product

P = ( u k 2 ) ( d c 2 )- f f g p . The change of sign takes place at

(3.13)

/3 = /3(")(k) with

mk4 /3(")(k) = -. ffg

(3.14)

As long as /3 < /3("), the product is positive and the two roots negative, mode k is stable. When p > /3("), it becomes negative and one of the roots is positive, the mode is unstable, convection sets in. The value /3 = /3(")(k) of the applied temperature gradient thus here defines the marginal stability condition that makes mode k neutral. One can observe that the negativity of the sum and the positivity of the product, the two stability factors, come from the stabilizing dissipative processes and that the instability factor involves a term in /3 arising from the advection of the temperature field. Let us come back to the marginal stability curve described by (3.14). It accounts for an increase of the marginal temperature gradient as k4 for k large, which expresses the growing efficiency of the stabilizing mechanisms as the scale of the fluctuations decreases (Fig. 3.2, left). If we trust in this relation, the longer the wavelength, the lower the threshold. However we should not conclude that the fluid layer is unstable at k = 0 for /3 = 0, i.e. AT = 0. As a matter of fact, this low-k behavior is an artifact of the one-dimensional model which neglects the z-dependence of the fluctuations and the associated dissipation processes: viscous damping by the horizontal component of the flow that closes the streamlines can no longer be neglected as k -+ 0 (Fig. 3.2, right), hence the corrected argument valid for k small. Let us keep v, as a reference since it is directly involved in the instability mechanism. We can estimate the order of magnitude of v, from the continuity equation a,v,

+ d,v,

= 0.

-

-

Boundary conditions on v, are a t the horizontal plates, a distance h apart, and imply a t-dependence such that dzv, v,/h and therefore ku, v,/h or v, v,/kh. But the presence of v, imposes us to take the x-component

-

80


Fig. 3.2 Left: For X = 2a/k > h the horizontal shear (gray arrows) becomes negligible while the vertical shear (black arrows) becomes dominant.

of the Navier-Stokes equation into account. We can simplify it as: - k p / p - vv,/h2

2~

0,

and introduce the so-evaluated pressure in the equation for v,, which yields:

atv, = -a,p/p + v (az2+ aZz)v, + a g e .

(3.15)

A sketchy analysis of the space dependence of the different perturbations then shows that, once expressed in terms of v, using the continuity equation, & p / p goes as vv,/k2h4, so that the pressure term dominates those involving v, on the r.h.s. of (3.15). As a matter of fact, it diverges as k-2 when k + 0, while the second term tends to zero as k2 and the third one does not vary with k. We thus arrive at an effective equation for v, replacing (3.3):

+

atv, = - ~ ( i / h ~ k ~ ) v a, g e , only valid in the limit k k,. Similar simplified analyses will be developed as exercises to study the effects of molecular diffusion on convection in binary mixtures (Exercise 3.3.3), the stability of an angular momentum stratification in a cylindrical shear experiment (Taylor-Couette instability, Exercise 3.3.5), or the emergence of spatial structures in reaction-diffusion systems (Turing instability, Exercise 3.3.4).

3.1.5

Vicinity of the threshold: linear stage

The study of linear dynamics of the fluctuations in the neighborhood of the threshold is best developed by first turning equations (3.3, 3.4) into dimensionless form. In order to do this, we have to choose length, time, and temperature scales. The thickness h of the layer is the obvious natural length scale. The thermal diffusion time over the distance h, 7 0 = h 2 / n , being retained as the time scale (the alternate possibility would be the viscous time T,,= h 2 / u ) ,the velocity scale then reads h/re = n / h . Since it is preferable to keep AT as the control parameter, the composite quantity3 rcu/crgh3 is taken as the temperature scale. Performing the changes x hx, t c-) rot ,... in (3.3, 3.4) we obtain

(3.17) (3.18)

A second dimensionless number has been introduced: (3.19) It is called the Prandtl number and characterizes the physical properties of the fluid, specifying which of the viscous diffusion (T,,= h 2 / u ) or the thermal diffusion (To defined above) is the dominant relaxation process. In gases P is of the order of unity and varies little with the nature of the gas since momentum (7,) is transported by the molecules themselves at the same rate as energy (To). In condensed fluids this number can largely vary. For example, in liquid metals (e.g. mercury) it is very small, typically < lo-’, since energy is efficiently transported by conduction electrons while atoms must be moved to smooth out velocity fluctuations, hence To > 1, the flow adjusts itself to the temperature field instantaneously, which can be understood from the consideration of equation (3.17) written as

P-latv,

N

o = axxvz+ 8 ,

showing that IJ, is merely obtained by integrating 8 over space. The dynamics is therefore simplified since we have just one relevant scalar field. On the contrary, when P is small, the inertia of the fluid cannot be neglected and a full hydrodynamic problem is recovered, with the vector nature of the velocity field and the incompressibility condition playing a crucial role. Let us stay in the limit P >> 1 and consider the critical mode ( v z ,8) sin(k,z) with k, N 7r ( r / h if the physical dimension is restored). Within the framework of the simplified model we get: N

and upon insertion in (3.18):

+ ~ 1 28 .)

8 + RV,= ( - 2

ate = 4

Dividing both members of this equation by n2 and defining: To

1

=I -

and

T

=

R - R,

-,

IT2

RC

(3.20)

with here R, = 7r4 (but this value is only anecdotal) we simply get:

Toate= r 8 .

(3.21)

The coefficient TO therefore presents itself as a characteristic evolution time ( T ~ / T ’ in physical units) for convection, while T measures the relative distance to the threshold and is of course our control parameter. Defining A as the amplitude of the most unstable convection mode and setting:

8 0; A ( t )sin(k,z) ,

(3.22)

84


we get from (3.21) the linear evolution equation for A

A =uA,

(3.23)

where u = T / T ~is its effective growth-rate. The corresponding time T = 1/u = T O / T therefore diverges as T-' close to the threshold (r 0.

Beyond threshold, for T > 0 ( R > R c ) ,several processes indeed come and limit the growth of A. First the dissipation increases, and second the destabilizing force decreases since part of the heat is transported by the flow, so that the bulk effective temperature gradient that governs the conductive part of the heat flux decreases below its nominal value ,B. It is of course possible to derive an accurate model of nonlinear evolution from the primitive equations in a systematic way. Here we rather continue to develop a heuristic formulation, guided by the result to be obtained. We no longer assume that P >> 1 but restrict ourselves to the consideration of the most unstable linear mode. On more general grounds than for (3.22), we then take {wz, B } = { V ( t )@ , ( t )sin(lc,z), } where V et 0 are two amplitudes functions of time. Injecting this assumption in (3.17, 3.18) we obtain:

v =P(0-2 V ) ,

(3.28)

O=RV-7r20.

(3.29)

Let us now try to complete (3.28, 3.29) with nonlinear terms arising from the advection of the fluctuations v V v and vV8. In the spirit of a first harmonic approximation now developed in space and not in time as for the van der Pol oscillator, p. 49, we guess that the terms that contribute are those resonating with the postulated dependence in sin(lc,z). From the continuity equation drvx dzwz = 0, assuming v, c( sin(k,z) one gets v, K cos(kcz), so that, in the equation for wz, the advection term v+d,v, w,dzv, varies as sin2(kcz) = f ( 1 - cos(2kCz)), i.e. produces nothing in resonance with sin(lc,z). Averaging over the thickness of the layer and over a wavelength, we thus expect a negligible contribution from these terms to (3.28) that remains unchanged at this order. The problem is different for (3.29). As a matter of fact, a parallel argument would also imply no complementary term, but this would not reflect the fact that, as indicated above, part of the heat is transported by the convection motion. The corresponding flux is easily identified t o the product w,8. As discussed in Sec. 3.1.2, below threshold, the destabilizing part of the convection mechanism relies on the advection, due to differential

+

+

4Notation g introduced to measure the intensity of nonlinear couplings is traditional. In the context of convection it should not be mistaken with the gravitational acceleration, but the risk is limited.

Instabilities. Chaos and Turbulence

88

buoyancy, of temperature fluctuations in a purely conductive temperature gradient. Above threshold, since part of the heat is transported by convection, differential buoyancy has t o be appreciated with respect to a conductive temperature gradient Peff which is decreased by the contribution of convection from its nominal value P evaluated from the applied temperature. That contribution, w,B c( sin2(k,x), produces (i) a second harmonic component c( cos(2kCx)that averages to zero over a fluctuation wavelength but also (ii) a term at k = 0 that precisely corresponds to the correction to the averaged temperature profile. Let us consider this correction as a variable in itself, call it P and look for its governing equation. A simple calculation then yields:

\ii = V O - bP,

(3.30)

where the first term on the right hand side is the source term issued from space-independent part of w,B and the second term accounts for its diffusive relaxation according to the Fourier law, at a decay rate b that could be computed explicitly. On the other hand, the argument sketched above is implemented by subtracting the convective contribution P from the nominal Rayleigh number R t o form an effective Rayleigh number re^ replacing it in (3.29). This yields:

0 = (R- P ) V - 7r20.

(3.31)

Equations (3.28, 3.31, 3.30) generalize the linear model derived previously. They form the celebrated Lorenz model that played such an important role in the development of ideas about chaos since 1963, when Lorenz first derived it5 by truncating a Galerkin expansion of the hydrodynamic equations on a trigonometric function basis, and further pointed out its “non-periodic” properties (see Appendix B, sB.4.2, p. 373). Let us first show how this model allows us to recover the effective Landau equation (1.19) introduced in Chapter 1, extending (3.23) to the nonlinear regime. As noticed earlier, close to the threshold, the dynamics of the system is very slow. Its evolution rate is proportional t o r = ( R - R,)/R, 0 so that the bifurcation is supercritical. All this has already been presented in the introductory chapter, see Figure 1.4 p. 14, and will be extended to general systems involving a single stationary mode in Chapter 4, especially in Exercise 4.6.3, p. 169. We have previously stated without justification that the bifurcated solutions corresponding t o convection beyond threshold are stable. Let us show how this arises from (3.33) using tools introduced in Chapter 2. Setting A = A‘, we easily get the equation for perturbation A’ by substitution. After simplification we get

+a +

70A’

= -2rA‘,

(3.34)

so that A’ decays for T > 0. In fact this is valid only close enough to the threshold, before new instabilities have any chance to set in. The previous argument indeed only holds as long as R M R,. When R >> R,, the relaxation time of 0 and V towards their equilibrium values, derived from (3.34), infinite a t threshold, shortens as T increases and rapidly

90


becomes of the order of magnitude of !P’s relaxation time. Adiabatic elimination of the latter is then no longer legitimate: !P is less and less enslaved to 0 and V but on the contrary gains a status of genuine degree of freedom. Secondary instabilities and the transition to chaos could be studied from the Lorenz system but, as far as physical observations are concerned, results turn out to be unrealistic, due t o the simplifications made. Having recognized that this model remains an excellent test-bed for chaos ideas (see Appendix B, gB.4.2, for a first numerical approach), we now consider nonlinear convection from a concrete experimental point of view. 3.2.2

Transition t o turbulence of convection cells

The actual situation is indeed less transparent than that described by means of the simplified Lorenz model. But one fact remains: the effective dimension of the problem increases with R. Unfortunately, the physical mechanisms that destabilize the cellular structure to produce of the secondary modes are much less intuitive than the primary mechanism. The method is the same in principle but, a t steady state, the base flow beyond threshold is now made of finite-amplitude time-independent convection cells. The study is considerably more complicated than when we had to deal with the uniform conducting state since the new base flow is periodic along one horizontal direction. Accordingly, the operator obtained through linearization now explicitly depends on space, which forbids the direct recourse to Fourier transforms to solve the problem. This will be reexamined theoretically later. For the moment, let us describe the cascade towards turbulence from a phenomenological point of view. The convection threshold was independent of the Prandtl number P , whose value just played some role in the nature of the primary mode, thermal when P >> 1, hydrodynamic when P > 1 (e.g., with highly viscous oils), the temperature field drives everything, inducing the vertical velocity component directly and the horizontal component indirectly through the continuity condition. Secondary instabilities specific to this case remain localized within thermal boundary layers close to the horizontal plates. These boundary layers get thinner and thinner as the Rayleigh number is increased and, a t some point, they become unstable against the plain Rayleigh mechanism. A stationary secondary instability called bimodal sets in, with rolls oriented at right angles with the primary rolls and located in the thermal boundary layers. Since the fluctuations are now modulated in the three directions of space, the regime is labelled ‘time-independent 3D’ in Figure 3.5. Time dependence next manifests itself as a periodic break-down and reformation of thermal boundary layers first analyzed by Howard. Strict periodicity is then lost and an irregular dynamics sets in.

92

3.2.2.2

Instabilities, Chaos and n r b u l e n c e

Intermediate and low Prandtl number fluids

At smaller P , the situation is more confused. Busse and his collaborators have identified a large number of possible secondary modes leading to a complicated picture in the (R, P, lc) parameter space called the Busse balloon7 owing to the global shape of the region where straight rolls are stable. When P 1 (water, gases) or smaller (liquid metals), the velocity field becomes dominant through specific contributions of the advection term v . Vv. The unstable secondary modes appear close t o the convection threshold and occupy the whole thickness of the layer. Cells enter a kind of free-wheel regime where friction on the plates is dealt with inside thin viscous boundary layers. Time dependence enters very early in the form of ‘Busse oscillations’ that are sorts of waves propagating along the convection rolls and due to an inertial call-back of roll axis undulations. Most often it turns out to be difficult to identify a range of Rayleigh numbers over which the periodic behavior is strictly regular, and the flow is often considered turbulent right at the onset of oscillations.

-

3.2.2.3

Transition towards turbulence, conceptual problems

At least for P >> 1there seems to be a small number of well defined steps on the way between the conduction regime and turbulence. This apparently supports the viewpoint advanced by Ruelle et Takens in 1971’ according to whom the stochastic behavior, a fundamental property of turbulence, generically appears at the end of a short cascade of three or four bifurcations. Previously, Landaug explained his understanding of turbulence as the result of an indefinite superposition of modes, each with its own time-space scale, i.e. quasi-periodicity with an infinite number of incommensurate frequencies. These two interpretations are sketched in Figure 3.5. In the context of experiments reported above (Fig. 3.4), neither the Ruelle-Takens interpretation nor a fortiori that of Landau, are satisfactory. All observations were made in containers that were very wide (in order to check theories developed for a laterally unbounded system). The 7For a n early review, see, e.g. F . H . Busse, “Transition t o turbulence in thermal convection” in Convective transport and instability phenomena, J. Zierep & H . Oertel Jr., eds. (Braun, Karlsruhe, 1982). Ruelle and F . Takens, “On the nature of turbulence,” Commun. Math. Phys. 20 (1971) 167-192. Addendum 23 (1971) 343-344. gL.D. Landau, “On the problem of turbulence,” Akad. Nauk. Doklady 44 (1944) 339, translation in Collected Papers 0 f L . D . Landau, D. ter Haar ed. (Pergamon Press, 1965), pp. 387-391.


‘3

.* +

pe‘riodic

periodic

2-periodic

2-periodic

93

.....

n-periodic

chaotic

Fig. 3.5 The “nature of turbulence” according to Landau (top) and Ruelle & Takens (bottom). Ellipses attached one to the next in a string feature quasi-periodic behavior as results from the superposition of periodic motions with incommensurate frequencies. The chaotic end of the Ruelle-Takens cascade is represented by a miniature of the attractor of the Curry & Yorke model that illustrates the disintegration of the ellipse born at the previous step (see Chapter 4, Exercise 4.6.13).

so-obtained convection patterns were rarely regularly organized but on the contrary presented lots of defects and inhomogeneities, so that the transition thresholds were not as sharply defined as a bifurcation point. Moreover a slow residual time dependence was often observed. Having recognized that these interpretation problems were mostly due to spatial disorder, which in turn resulted from the presence of a large number of cells, and that lateral boundaries at large distances were ineffective in maintaining long range order in the patterns, experimentalists have tried to better control the situation by turning t o systems with a small number of cells, hence lateral dimensions of containers of basically the same order of magnitude as their heights. Confinement effects can be appreciated through aspect ratios defined as

r = e/h,

(3.35)

where e represents the typical lateral extension of the system, see Figure 3.6. We shall reexamine their physical role later in Chapter 4, $4.1. Early experiments reported above were performed in the limit r >> 1 that characterize extended systems and for which the concept of spatiotemporal chaos to be introduced in Chapter 5 seems more appropriate.


94

1 2

h

h

Fig. 3.6 Aspect ratio for closed systems, either confined (left) or extended (right).

-

By contrast, strongly confined systems, characterized by I' 1, can be expected to better fit the framework proposed by Ruelle and Takens and their concept of temporal chaos. As a matter of fact, guaranteeing strong spatial coherence among a small number of convection cells, confinement effects should be instrumental in restricting the dynamics to couplings within a small set of effective variables.

3.2.3

Transition toward chaos in confined systems

The literature about the transition from regular to chaotic time behavior is sufficiently rich that we can limit ourselves to the presentation of few experimental results obtained at the beginning of the eighties as typical examples of the main scenarios. This sketchy description is given mainly as an illustration of the kind of phenomena to be understood theoretically rather than as a review that would rather be premature at this stage. Consult the general bibliography for more detailed information, especially [Hao (1990); CvitanoviC (1989)l.

3.2.3.1 Subharmonic cascade

-

The first experiment to be reported here has been performed by Libchaber et Maurer." Liquid helium with P 1 is placed in a parallelepipedic container with aspect ratios r2 = 2.4, rY = 1.2. At the beginning, stationary convection sets in beyond some threshold R,. At R N 30R, the system experiences a bifurcation toward an oscillatory regime. Then, at R = 39.5RC, a second mode with an incommensurate period sets in. This two-periodic ''A. Libchaber and J . Maurer, "Une expbrience de Rayleigh-BBnard en gbombtrie rbduite; multiplication, accrochage et dbmultiplication de frkquences," J . Physique Colloques 4 1 4 3 (1980) 51.


95

Time series of the temperature signal measured at a given point during the first steps of a subharmonic cascade after Libchaber and Maurer, Note 10.

Fig. 3.7

regime persists up to R N 40.5Rc when, while shifting, the second period gets locked to twice the first one, the system is then periodic with a period 2 T . The scenario under study now begins: a second period doubling (period 4T)at R N 42.7Rc (Figure 3.7). After several supplementary period doublings (period 8T,16T,. . . t h e system enters a chaotic regime for R > 43Rc. A complementary study of Fourier spectra would show first fine lines at one frequency and its harmonics, then a second family of lines and many combinations (two-periodic regime), then, after the locking, a return to a simpler spectrum with one fundamental line at w = 27r/T and its harmonics. The period doubling cascade manifests itself by the growth of subharmonics at w/2, next w/4, etc. As long as the system is periodic, no matter how long the period, the spectral lines remain narrow but when it becomes chaotic, they get measurably enlarged at their foot.

3.2.3.2

Chaos o n a two-periodic background

This transition, closely reminiscent of the scenario originally proposed by Ruelle and Takens, has been observed roughly at the same epoch by Dubois and Berg611 again in parallelepipedic geometry with similar aspect ratios, = 2, ry = 1.2, but this time with silicon oil ( P 21 130). By contrast with the previous experiment, visualization of the structure was possible by means of differential interferometry, Fig. 3.8 (top), which made easier the understanding of motions at the origin of the observed fluctuations and the choice of points where t o measure the velocity by LASER Doppler anemometry. Here is the (simplified) sequence observed: 1) conduction regime up t o Rc. 2) Time-independent convection from R, to R N 215Rc. 3) Bifurcation towards a periodic regime with period TI. 4) Two-periodic dynamics with a second period T2from R N 250R, up. Geographically well separated l l M . Dubois and P. BergB, “InstabilitBs de couche limite dans un fluide en convection: Bvolution vers la turbulence,” J. Physique 42 (1981) 167.


96

I,

w

t

Fig. 3.8 Top: Isotherms in silicon oil can be visualized by differential interferometry; the fringes originate from the variations of the refraction index induced by the local temperature gradients (picture kindly provided by M. Dubois). Bottom: Fourier spectra (left) and time series (right) of a velocity component at a given point in the experimental cell. In the two-periodic regime, the spectrum is essentially composed of two principal lines, each of which can be attributed to a definite fluid perturbation well localized in space. The chaotic regime is characterized by spectral lines with enlarged foot and a large amount of power at low frequency that corresponds to the slow fluctuations in the corresponding time series. After Dubois and BergB, Note 11.

the two oscillation modes are weakly coupled, which explains the relative robustness of the two-periodic regime and a characteristic alternation of locking/unlockings when the ratio of the periods, that slightly shifts with R, passes from incommensurate to commensurate values and vice versa. 5 ) For R > 305R,,temporal chaos enters as an irregular slow modulation of a locked periodic behavior that gives a series of widened spectral lines and low frequency power in the Fourier spectrum, Fig. 3.8 (bottom).


..........,.,,........., ,

turbulent burst

,

. . . .,. ,

97

.... laminar intermission

Fig. 3.9 The intermittency scenario: A periodic regime (“laminar” intermissions) is irregularly interrupted by chaotic bursts that become more frequent as R increases beyond the intermittency threshold. Upper trace: regular signal observed below threshold. Time is running from left to right. After Berg6 et al., Note 12.

3.2.3.3 I n t e r m i t t e n c y The third scenario to be described here has also been observed by Berg6 and Dubois” with the same fluid and the same experimental set-up but with a slightly different initial convection structure. Accordingly, a different transition scenario developed after a single step involving a secondary instability mechanism with a hot droplet transported by the general convection and playing the role of a pacemaker. Convection was time-independent up t o R = 25QR,, then periodic with period T . Not far above a subharmonic bifurcation (hence period 2T), for R = 29QR,, the system experienced a transition to chaos with irregularly distributed “turbulent” bursts interrupting the previously observed regular periodic behavior forming “laminar” intermissions. When the Rayleigh number was increased, the frequency of the bursts was seen to increase. Time series of the velocity signal before and after the transition are displayed in Fig. 3.9, upper trace and lower trace, respectively. 3.2.4

Dynamics of “textures” in extended s y s t e m s

By contrast with what has just been described, before 1975 the transition to turbulence was studied in extended systems and focused more on the occurrence of a developed turbulent regime where most of the spatial structure was lost and the time dependence strongly irregular. They did not recognize the often observed low frequency noise as an interesting phenomenon related to the transition process. The study of the emergence of chaos in confined systems has also led to reconsidering the situation for extended lZP.Be@, M. Dubois, P. M., Y. Pomeau, “Intermittency in Rayleigh-BBnard convection,” J. Physique Lettres 40 (1979) L505.

98


Fig. 3.10 Texture observed in convection at large Prandtl number as seen from above. Picture kindly provided by V. Croquette, Note 13.

systems and introducing the notion of spatio-temporal chaos as an element of interpretation of the transition process. In practice, the instability mechanisms preserve coherence at the local scale (few convection cells) but are unable t o maintain it on a global scale (the set-up). This can be understood as the result of the possible interference of a large number of neighboring modes easily excited immediately beyond threshold ($3.1.5and Chapter. 5). In the absence of any induction process forcing the growth of regularly oriented “clean” convection structures, disordered patterns with many defects of all sorts, called textures, are generally observed. Here again we have to distinguish between fluids according t o their Prandtl number. 3.2.4.1 Textures in fluids with high Prandtl number

When P is large, quasi-stationary convection structures are obtained, which relates t o the fact that the dominant field is the temperature, a scalar, and that the fluid layer mostly behaves as a gradient system (see Chapter 2, $2.1.3, p. 33). It is then possible to interpret the principal features of the textures observed in terms of a single amplitude field. An example is given in Fig. 3.10, where one can easily identify grains of convection rolls with nearly uniform orientations, grain boundaries along which two grains with different orientations meet, dislocations13 where a pair of rolls suddenly ends. It should also be noted that rolls arrive mostly perpendicular t o the lateral boundaries and that the frustration implied by this topological constraint is partly resolved by the presence of a large scale curvature of the rolls. The evolution of such textures is very slow, when compared t o the l 3 A . Pocheau & V. Croquette, “Dislocation motion: a wavenumber selection mechanism in Rayleigh-BBnard convection,” J . Physique 45 (1984) 35-48.


99

Fig. 3.11 Weak turbulence observed in a cylindrical cell with r = D / h = 12 in liquid helium at low Prandtl number. Typical background noise is the gray line in each figure. For R = 3.35Rc low frequency noise develops first. At R = 3.70Rc noisy Busse oscillations at finite frequency around 0.5 Hs have settled in the system. As seen for R = 4.50Rc and beyond, the level of noise continues to increase gently. After Libchaber and Maurer. Note 14.

velocity of the convecting fluid. The transition from 2D t o 3D convection is mediated by the defects and the emergence of (weak) turbulence can be understood as a kind of “melting” of the global pattern, with progressive loss of local order. 3.2.4.2 Transition at small Prandtl number and large aspect ratios

All experimental observations show that turbulence occurs early when P is small. This feature has to be attributed t o the fact that the viscosity is low and that Reynolds numbers constructed from the velocity induced by convection and the size of the cells rapidly become large. Like for lowdimensional systems, the complexity of the dynamics is considerably enriched by inertial effects that favor oscillatory behavior (here mainly the Busse oscillations). This explains that mode interactions, even close to the convection threshold, generate a much more ‘Lactive”behavior than what is observed at higher Prandtl numbers. Another source of complexity comes from the existence of large scale flows directly generated by curvature and defects in the global texture as now shown. Figure 3.11 illustrates the scenario observed again by Libchaber and Maurer14 in liquid helium but in a cylindrical container with diameter D and aspect ratio l? = D / h = 24. Remarkably enough, a low frequency noise sets in before the trace of any secondary instabilityin the form of Busse oscillations is apparent, and when the latter develops the system is already disordered so that the corresponding frequency is not sharply 14A. Libchaber and J. Maurer, “Local probe in a Rayleigh-BBnard experiment in liquid helium,” J. Physique Lettres 39 (1978) L-69.

100


Fig. 3.12 Cyclic evolution from state (1) --f (2) -+ . . . --f (6) + (1): Mechanism of nucleation-dissociation-migration-annihilation of dislocations pointed by Pocheau et al, Note 15, and here illustrated with pictures kindly provided by V. Croquette from a similar experiment with r = D/h = 40).

defined. It took some time before this behavior could be understood. In an experiment where helium was replaced by argon under pressure at room temperature, making visualizations possible, Pocheau et aZ.I5 later showed that the noise developing slightly above threshold was due to the synchronization loss of an initially periodic process of nucleation, migration, annihilation of dislocations, as illustrated in Figure 3.12. The dislocation migration was driven by a secondary flow at the scale of the container, the existence of which was predicted by earlier theoretical studies16 and explicitly confirmed by specific experiments later.17 Still for low Prandtl numbers, somewhat above the range of Rayleigh numbers where the convection pattern is made of possibly slowly evolving but essentially straight rolls, a much more disorganized active state is observed in the form of rotating spirals as illustrated in Figure 3.13. This 15A. Pocheau, V. Croquette, and P. Le Gal, “Turbulence in a cylindrical container of Argon near threshold of convection,” Phys. Rev. Lett. 55 (1985) 1094-1097, later reviewed by V. Croquette, “Convective Pattern Dynamics at Low Prandtl Number. Part I, 11,” Contemporary Physics 30 (1989) 113-133, 153-171. lB(a)E.D. Siggia, A. Zippelius, “Pattern selection in Rayleigh-Bbnard convection near threshold,” Phys. Rev. Lett. 47 (1981) 835-838. (b) P. M. and J.M. Piquemal, “Transverse phase diffusion in Rayleigh-Bbnard convection,” J . Physique Lettres 43 (1982) L253-L258. (c) M.C. Cross, A.C. Newell, “Convection patterns in large aspect ratio systems,” Physica D 10 (1984) 299-328. 17V. Croquette, P. Le Gal, A. Pocheau, P. Guglielmetti, “Large-scale flow characterization in a Rayleigh-Bbnard convective pattern,” Europhys. Lett. 1 (1986) 393-399.


101

Convection in C02 with I’ = D / h N 150, spiral defect chaos observed at r = 0.536 (left) and r = 0.894 (right). After Morris et al., Note 18, courtesy G. Ahlers Fig. 3.13 (UCSB).

spiral defect chaos, observed in particular by Morris et aL1* permanently evolves in both space and time. It takes place only a t large aspect ratios but then does not depend on the lateral shape of the container. Furthermore, it is extensive in the sense that it can be characterized by a surface density of spiral cores seen to increase with increasing Rayleigh numbers beyond a threshold that depends on the Prandtl number. Curvature-induced secondary flows seem essential to its occurrence.

3.2.5

Turbulent convection

The local study of convection structures has, since its very beginning, been completed by measurements of the heat flux through the whole experimental container, characterizing the global behavior. Results are usually expressed in terms of the dimensionless Nusselt number:

N =

total heat flux conduction heat flux

where the ‘total heat flux’ is the quantity actually measured and the ‘conduction heat flux’ is the flux that would be computed from the temperature difference upon assuming that the fluid is at rest in the pure conduction state. Hence one gets N 1 when R < R,, while N - 1 measures the 18S.W. Morris, E. Bodenschatz, D.S. Cannell, G. Ahlers, “The spatio-temporal structure of spiral-defect chaos,” Physica D 97 (1996) 164-179.

102

Instabilities, Chaos and n r b u k n c e

contribution of convection. Close to threshold, one expects:

R - Rc N-lKVz6cX-,

RC

(3.36)

since both 9 and v, vary as [(R- Rc)/Rc]l/z,which is indeed well observed experimentally, see Figure 3.14. Far beyond threshold, from scaling arguments familiar in the theory of turbulence (Chapter 7), a power law behavior is expected instead:

N

-

R’.

Early experiments seemed to support a theory by Malkus predicting

y = 1/3 but the range of Rayleigh number studied was too narrow, while other studies for P 0 and conclude that the system bifurcates towards a cellular structure at T = 0. Show in particular that the most dangerous modes correspond to Ikl = 1. c) Sketch the marginal stability surface (s(kr) = 0) in the threedimensional space (k,, ky,r); determine the set of marginal modes and the domain of unstable wavevectors s(k,T) > 0 when T > 0. [Answer: Figure 3.15 (left)]. d) System (3.37) is isotropic in the (z-y) plane. Consider now the modified anisotropic model

atw= T w - [(a,, + q2- a,,]

w.

(3.38)

Sketch the marginal stability surface for this case and conclude that, by contrast with the isotropic case, the linear stability operator selects a non degenerated mode. [Answer: Figure 3.15 (right)].

Fig. 3.15 Marginal stability surfaces for the linearized Swift-Hohenberg model in the isotropic (left) and anisotropic (right) cases.

3. Life and Death of Dissipative

105

StTUCtUTeS

2) Consider now the same system but restricted to one space dimension:

a,v = TV

-

(axx+ q 2 v ,

(3.39)

for a function v(x, t ) defined on a finite interval of length e with boundary conditions v = dxxv = 0 at x = 0 and x = e. Check that the eigenmodes can be taken in the form V = Asin(k,x) with Ic, = n.11. Determine the marginal stability condition T = rnm) ( for mode n at given C, and next the instability threshold rc(C) = inf, (e), as a function of e. How does the spatial resonance between the intrinsic length-scale A, = 2n/kc = 2. and the size e of the system manifest itself? Find the condition for two neighboring modes being simultaneously marginal. The student is encouraged to perform the same (but much more difficult) study for boundary conditions v = a x v = 0 at x = fC/2. This is more typical of the general case since simple trigonometric lines are no longer appropriate. The eigenmodes will be searched for as superpositions of elementary solutions to the 4th order differential equation

(el rim

sv = [r -

($

+ q2]v

that fulfill the boundary conditions (the scalar s is the eigenvalue). Separate odd from even solutions and find corresponding marginal conditions (given by transcendental equations to be solved numerically by some root-finder program.) This model, completed by appropriate nonlinear terms, will be used again in Chapter 4, Exercise 4.6.2 and in hands-on numerical experiments, Appendix B.

3.3.2

Rayleigh-Be'nard convection: detailed study

The purpose of the exercise is to go beyond the semi-quantitative approach developed in 53.1.4 and determine the marginal stability condition from the full primitive equations. The theory rests on the Boussinesq approximation of moderate heating which supports the idea that the fluid's physical parameters are independent of the temperature, except the density in the term responsible for the differential buoyancy force. The linearized thermo-hydrodynamic equations governing two-dimensional (2, z ) pertur-


106

bations then readz1:

&(azz+ a,,)~,= P ((azz+ a z z ) 2+~ ,a d ) , ate = R W ,

+ (azz+ a,,)s.

(3.40) (3.41)

These equations are written here in dimensionless form after elimination of the pressure and horizontal velocity component. The scales are the same as those leading to (3.17, 3.18). As far as the (z,t)-dependence is considered, their structure is also the same as that of the simplified system, but they now retain the additional z-dependence of the fluctuations explicitly. Boundary conditions are set at the position of the plates zp, at top, zt = 1 or +1/2, and bottom Z b = 0 or -112, one or the other choice making computations more transparent depending on the cases considered. We consider here infinitely good heat-conducting horizontal plates. Accordingly the temperature is strictly fixed by heat baths so that the fluctuations are zero there:

eltp = 0 .

(3.42)

For the velocity components, rigid plates imply a no slip condition %lZP

= at%lzp = 0 .

(3.43)

In practice, calculations are easier with the somewhat artificial stress-free conditions considered initially by Rayleigh, which leads to the replacement of (3.43) by = ~ Z Z ~ , l Z p= 0 .

(3.44)

In the following we consider symmetrical cases where top and bottom conditions are identical but any non-symmetrical condition (good/bad conductor, no-slip/stress-free) can be considered in the same way at the expense of a (much) more cumbersome analysis since parity considerations are no longer useful to classify the solutions.

1) Normal mode analysis in the stress-free case (Rayleigh solution, 1916). Boundary conditions are set at zp = 0 and 1. a) For a laterally unbounded layer, check that solutions to (3.40, 3.41) can be taken in the form:

(w,,0) = ( V ,0 )sin(nm) exp(ikz) exp(s,t) . 'lSee p. 108 for a derivation of equations and boundary conditions.

3. Life and Death of Dissipative Strvctures

107

b) Determine the marginal stability condition for the mode n that first bifurcates when R is increased. Find the corresponding critical wavevector and threshold. [Answer: (3.45) k, = 7 r / f i , R, = 277r4/4. See Fig. 3.16, curve A.] c) Show that the linear dynamics close to the threshold is governed by: 70s 1T - [;(k - kc)’,

( R - R,)/R, and coefficients TO and 50 to be computed. 2) Normal mode analysis in the no-slip case. The exact solution, obtained by Pellew and Southwell (1940) is presented in [Chandrasekhar (1961)l. The corresponding marginal stability curve is given in Figure 3.16, as line B’ (fine, solid) with critical conditions k, x 3.11632, R, x 1707.76. Here we look for an approximate solution by a so-called Galerkin method, a special case of weighted-residual approximation introduced, e.g. in [Finlayson (1972)l. In a few words, the solution, expanded on a complete basis of functions, is further injected in the equations that are projected on a complementary basis using some scalar product. The Galerkin method comes in when the functions in the original and complementary bases are identical and fulfill the boundary conditions of the problem. The approximate solution is seen to converge to the exact solution as the number of functions is increased, especially when the problem has an underlying variational structure, which is the case for Rayleigh-BCnard convection, as discussed at length in [Chandrasekhar (1961)l. with

T

Here, the solution of (3.40, 3.41) is searched for in the form:

{ v r ( z , z , t ) , e ( z , z , t )= } { V ( z ) ,O(z)}sin(k,z) exp(st) . The z dependence of V, and 0 is taken as polynomials. Boundary conditions are set at z = f 1 / 2 . a) Show that in order to fulfill boundary conditions automatically, one

must take:

V ( z )= O(2)=

(a - 2 ) 2 P V ( Z ) , (i- z”)Po(z),


108

where P,,(z) and Pe(z) are polynomials in z , P,,(z) = C,"==, Vnzn, Pe(z) = C,"=,Onzn. The approximation enters when polynomials are truncated beyond some given maximal degree N. b) The differential problem being formally written as CU = 0, where U has two components 6 et the projection onto the basis is defined by integrals

v',

l)12zn($ -z

~ ) ~ ( C Udz) ,=~0 ,

- z2)(CU)edz = 0 ,

+

for n = 0 , 1 , . . . ,N . This leads to a system of 2 ( N 1) linear equations for the 2 ( N 1) unknown coefficients introduced in the polynomials. Considering only the stationary case a t marginality, i.e. s = 0 , derive the system at lowest significant order, i.e. N = 0 (2 equations for 2 unknowns, 00and VO)and the corresponding marginal stability condition; compare the result to the Rayleigh solution (3.45). Then compute the critical wavevector and the threshold; further compare them t o the exact result given above.

+

[Answer:

+

+

28 (k4 24k2 504)(k2 R(")(k) = 27 k2

+ 10) 7

(3.46)

threshold: k, x 3.1165 Y exact value, R, x 1750 2.5% too high only. See Fig. 3.16, curve B (dot-dashed line), but so close an agreement is somehow accidental!] Derivation of system (3.40-3.44). In the two-dimensional case (2,z ) , the continuity, Navier-Stokes, and Fourier equations read:

The formally quadratic terms have been dropped owing t o the linearization step. Following the usual procedure, pressure is eliminated by differentiating the equation for v, with respect to x, the equation for vx with respect

109


500 -

k Fig. 3.16 Marginal stability curves of the most dangerous modes in the stress-free case (curve A) and no-slip case (curves B). Thin solid line 4 exact result. Dot-dashed line + Galerkin approximation at lowest significant order.

to z and subtracting the two. In order to eliminate w,, the result is further differentiated with respect to x and the so-obtained d,w, replaced with -&w, using the continuity equation. This leads to:

&(a,, + & z ) V t

=V

( L

ate = K ( & ,

+ & z ) 2 w , + ag&,8 + &,)e + Pv,

1

1

which is finally cast into (3.40, 3.41) by scaling v, and 8 appropriately. No-slip and stress-free boundary conditions respectively read w, (zp) = w,(zp) = 0 and d,w,(zp) = w,(zp) = 0. Differentiating the conditions on w, with respect to x and replacing dzwz with -a,w, yields the boundary conditions (3.43, 3.44), expressed in terms of w, exclusively.

3.3.3

Simplified model of convection in a binary mixture

Thermal convection in the presence of additional molecular diffusion processes is called thermohaline, by reference to the diffusion of saIt in water. Here we consider the emergence of convection in a binary mixture. The local state of the fluid is thus characterized by the concentration C of a solute, in addition to its temperature T and its velocity v. The temperature gradient is still generated by the temperature difference AT, while a concentration gradient is applied by putting the fluid layer in contact with two reservoirs at different concentration through appropriate porous


110

membranes. The solute concentration difference between top and bottom is a second control parameter and molecular diffusion (Fick law, coefficient D ) is a supplementary stabilizing mechanism, while solute advection may contribute to the destabilization of the layer. We develop a simplified onedimensional model in the spirit of Section 3.1.2, i.e. assuming fluctuations that are functions of x and t only, while the driving gradients are imposed along the z direction. 1 ) Construction of the model: The fluid layer is supposed to be at rest (v 0) in contact with two baths at temperatures Tt and Tb = Tt AT (heating from below implies AT > 0) and concentrations C, and Cb = C, AC (AC positive or negative). The purely diffusive temperature and concentration profiles read:

+

T ( z ) = Tb - p z

and

+

C ( z ) = cb - p ' z ,

where ,B = A T / h et p' = AC/h are the applied gradients. The differential buoyancy force is still induced by variations of the density p but it now has two origins, thermal expansion and composition change. Accordingly, the state equation can be taken as

where a (> 0) is the same coefficient as that introduced previously and where the sign of a' depends on the composition of the mixture. Justify the equations governing the linearized model:

and explain the origin of terms pv, and p'v, and discuss in simple terms the possible instability mechanisms involving each fluctuation. Take h as length unit, re = h2/K as time unit, introduce the Lewis number L = D/K in addition to the Prandtl number P = n/u,define the control parameters

R=

agATh3 KU

et

R'=

a'gACh3 ~

Du

(thermal and chemical Rayleigh numbers), and finally cast these equations


111

in the form

atv, = P(a,zv,

+ 8 + Lc) ,

ate = ax2e+ RV, ,

atc = Lazzc+ R ' ~ , .

Interpret the Lewis number from a physical viewpoint; what can be its order of magnitude in a liquid, in a gas? 2) Normal mode analysis: a) Introducing {v,, 8, c } = {V,0,C} cos(kz) exp(st), write down the algebraic linear system fulfilled by the amplitudes {V,0,C}. (Here we assume directly and without justification that k T (ie., in dimensional units X = 2h where X is the wavelength of the unstable mode). b) Show that for highly viscous fluids ( P + m), the resulting system of three equations for three unknowns can be reduced t o a system of two equations for two unknowns by eliminating the velocity component v,. In the following we keep this supplementary assumption but the general case can be treated in the same way using Exercise 2.5.3, Chapter 2, p. 63. c) Derive the compatibility condition of the simplified two-dimensional system and show that, by contrast with ordinary convection, complex roots are possible in the marginal case.

-

3 ) Different instability modes: a) From a discussion of the sign of coefficients of the quadratic equation expressing the compatibility condition above, find the threshold of the stationary instability mode (the product of roots change its sign), then that of the oscillatory mode (the sum of roots change its sign). b) In the parameter plane of parameters R' (horizontal axis) and R, draw the graph of these threshold conditions. Discuss the nature of the regime expected in each of the regions bound by these lines. Find the coordinates of the point where the system is simultaneously marginal against the two modes. Try to explain the physical origin of the oscillations by returning to the different contributions to the density changes and their respective relaxation times. [Answer: Figure 3.17, next page. The mechanism result from an interplay of differential buoyancy with two competing dissipation processes, one (thermal diffusion) being much faster than the other (molecular diffusion). This induces delays and phase shifts between the different fluctuations, ending in overshoots and oscillations.]


112

conduction regime L

714

Fig. 3.17 Stability diagram for convection in a binary fluid mixture. When R’ > 0, the temperature and concentration fluctuations cooperate, whereas when R’ < 0, they play antagonistic roles. Oscillations occur as their evolutions get sufficiently out of phase. At the intersection of the two lines (called a codimension-two point, where two parameters have to be tuned) the two mechanisms are equally efficient to destabilize the layer.

3.3.4

Turing patterns and reaction-diffusion systems.

In chemistry, mechanisms combining reaction and diffusion may produce dissipative structures called Turing patterns. Here we consider a simplified reaction-diffusion system where two species U and V with diffusion coefficients Du and D V also react with each other. In dimension one, with coordinate x, the model reads:

a,u = F ~ ( uv) , + Duax,u, a,v = F ~ ( uv) , +Dvaxxv,

(3.47) (3.48)

where reaction terms Fu and FV need not be specified at this stage. 1) Neglect diffusion (Du= Dv 0) and assume that a fixed point solution (Uo,Vo)exists. Linearize the governing equations around that point, set

a = auFuio,

b = avFuio,

c = auFvio,

d = avFvio,

computed at (UO, VO), next determine the conditions on a, b, c, d that guarantee the stability of solution (UO,VO). [Answer: the linear stability matrix must have eigenvalues either real and negative or complex with negative real parts, hence negative sum ( a d < 0) and positive product (ad - bc > O)]. ‘10’meaning

+


113

2) Assume that these conditions are fulfilled and add the effect of diffusion ( D u # 0, DV # 0 ) . Determine the dispersion relation of fluctuations around the state (VO, VO)supposed uniform in space. Write down the linearized system governing the amplitudes and V ( t ) )of Fourier normal modes in exp(iIcz) and show that the instability, if any, is necessarily cellular and stationary (Ic # 0, w = 0 ) . Show that the occurrence of the instability requires:

v(t)

aDv

+ dDu > 0

and

4(ad - bc)DuDv 5 ( a D v

+ dDu)2,

in addition to the conditions already found. When this is the case, determine the range of unstable wavevectors Ic. 3.3.5

Taylor-Couette instability

We study the stability of the flow between infinite coaxial cylinders rotating at different angular speeds (Couette flow). This problem was studied by Rayleigh (1916) at the limit of zero viscosity. Taylor (1923) developed the first theoretical analysis of the viscous case and performed the corresponding experiments. In the geometry of Figure 3.18 (left), assume that the base flow is purely azimuthal, show that wd = T W O ( T ) with WO(T)

=a

+ b/r2,

(3.49)

obtain a and b from the no-slip condition W ( T I ) = wi at ~ i i ,= 1,2. [Answer: a = ( W 2 T i - W i T : ) / ( T ; - T ; ) , b = ( W 1 - W 2 ) T : T s ) / ( T ; - T : ) . ]

1) Rayleigh instability mechanism (see also [Chandrasekhar (1961)] or [Drazin and Reid (198l)l). The base flow is characterized by the fact that the centrifugal force at distance T from the axis, ~ T W ~ ( T is) , compensated by a centripetal pressure gradient. Consider a rotating fluid particle displaced from distance T t o distance T 6~ > T . In the absence of viscous friction, angular momentum P T W ~is a conserved quantity. From this conservation law, derive its angular speed a t the new position and compare it with that of a fluid particle originally at the same place.

+

2) If the speed of the displaced particle is smaller than that of the surrounding fluid, the local pressure gradient is larger and pushes the particle back to its original position, the purely azimuthal flow is stable. In the opposite

114

Instabilities, Chaos and lbTbuknCe

tve

8 Taylor-Couette

4

Ganler

Dean

Fig. 3.18 Left: Geometry of the Taylor-Couette experiment. Right: Examples of curved flows.

situation it is unstable. Show that this stability condition reads dw dr

-6r

W

2 -2- br , T

then turn it to the form:

6 (r”)

20

and express this Rayleigh stability criterion with words.

3) Application to the Couette profile and other curved flows. a) Coming back t o the base flow profile (3.49) identify the different possible cases (rotation direction identical or different, and when the rotation directions are identical, which cylinder is rotating faster) and find the situations that are stable according to the Rayleigh criterion. b) When cylinders rotate in opposite directions, determine the region which is stable according to Rayleigh. c) By comparison with the case of convection, and by anticipation of Chapter 6 about shear flows, guess the role of viscosity, especially when the flow is mechanically stable. d) The same instability mechanism is expected to work when flow lines are curved. Identify the regions of the flow where the centrifugal instability can develop according to Rayleigh, in the boundary layer flow along a concave wall (Gortler instability; what about a convex wall?) or in the flow along a curved channel (Dean instability) depicted in Figure. 3.18 (right).

Chapter 4

Nonlinear Dynamics: from Simple to Complex

We now examine in more detail the theoretical context where the convection experiments described in the previous chapter can be best situated. The interest of this study stems from the fact that most instabilities that develop in bounded continuous media enter the framework of dynamical systems theory. Its foundation rests on an analysis of the effects of lateral confinement, $4.1. In a strongly confined system, even far enough from the instability threshold, the space dependence of unstable modes remains frozen and the state of the system is sufficiently well characterized by few amplitudes playing the role of effective degrees of freedom, which permits the analysis of the transition to turbulence in the spirit of Chapter 2. Here we limit ourselves to a heuristic presentation of the recent developments of nonlinear dynamics without insisting much on mathematical aspects. We are thus mainly interested in characterizing sustained regimes of dissipative dynamical systems obtained after the decay of transients. These asymptotic regimes reached in the limit t + f o o are represented in phase space by objects called limit sets. Stable limit sets are reached in the normal course of time, i.e. for t + +m and unstable limit set in the reverse case t + -oo. We have already seen examples of limit sets, fixed points and limit cycles for time-independent and periodic regimes, respectively. More complicated, multi-periodic and chaotic, behaviors are accounted for by objects called limit tori and strange attractors, which remain to be introduced. The second main purpose of this chapter is the description of the growth of complexity observed as the number of instability modes increases when control parameters are varied. The transitions from timeindependent states to regular periodic or multi-periodic regimes, next to chaos, take place according to universal scenarios that are independent of the particular system studied. Section 4.2 is devoted to the simplest ones. 115

116


By contrast with regular regimes, chaotic states are characterized by their unpredictability, generally associated with a fractal occupation of phase space. These different facets are introduced at a theoretical level in Section 4.3. Next we turn t o empirical approaches useful to interpret experimental results in Section 4.4. However, all this is valid only in so far as the projection of the dynamics onto well isolated spatially frozen modes makes sense. In practice, unfreezing takes place when lateral confinement effects are not strong enough to maintain coherence much beyond the cell size. In systems with many cells chaos then develops both in time and in space, but this topic is defered to Chapter 5. 4.1 4.1.1

Reduction of the Number of Degrees of Freedom Role of aspect ratios

Let us come back to eigenmodes obtained from the linear stability analysis of the instability studied, e.g. convection. The problem has been formalized in Section 3.1.3, p. 75. We assume that confinement is specified by lateral boundary conditions at a distance t , whereas the instability generates cells with typical scale A,. The aspect ratio can be defined as I’ = !/A,. The difference between the cases ‘T 1”and ‘T>> 1” is illustrated in Figure 4.1 and 4.2. Exercises 4.6.1 and 4.6.2 should be worked out t o gain a more analytical understanding of the problem. We first consider the case when the horizontal dimensions are of the order of the wavelength A, so that all modes are physically distinct (Fig. 4.1). Only the value of the corresponding thresholds R I , Rz,. . . matters, the two-dimensional presentation of the different modes is given just in order t o rationalize the classification in terms of the number of substructures in x and z , which is supported by the idea that these states should stay close to the branches that would exist in the absence of lateral boundary conditions (fine solid lines). They account for a kind of spatial resonance between the lateral extension of the system and the length scale originating from the instability mechanism, ending in the formation of an integer number of roughly circular cells, as already illustrated in Figure 3.8 (top), p. 96 (visualization of convection in silicon oil). The lower branch would then correspond to the “fundamental” states with one layer of cells and the upper one to “excited” states with two layers of cells stacked one above the other as could be obtained at higher values of the control parameter. N

4.

117

Nonlinear Dynamics: from Simple to Complez

A

R5

Fig. 4.1 R1 < R

R

R3

----

R2

----

Spectrum of the linearized stability problem for a confined system, 1 is unstable, the others are stable.

< Rz,mode

r

N

1; for

In practice, the actual structure of the modes can be obtained at the price of a full three-dimensional calculation, in general difficult and with little supplementary insight into their physical significance. Since the spectrum of the linear stability problem is entirely made of isolated values far from one another, it is in principle easy to limit the number of linearly excited modes ( R > R, hence u, > 0) from those that are still damped ( R < R,, u, < 0, IUml >> 1). By contrast, when I? >> 1, that is C >> A, (Figure 4.2), the spectrum is quasi-continuous. It is obtained from the dispersion relation for a laterally unbounded system by adding a quantization condition on the wavevector, typically k, rn/C for n cells of width X/2 over a interval of width C. The difference between neighboring wavevectors for patterns with f1 cell is k,+1 - k, = r/C > 1; for R > R, the modes belonging to a range O ( d m ) centered at kc are unstable and may serve to build the nonlinear pattern.

4.1.2

Low dimensional effective dynamics

4.1.2.1 Framework

From a general viewpoint, it may seem reasonable t o take advantage of the work done in solving the linear stability problem t o treat the nonlinear problem. This is done by expanding the solution onto the basis of eigenmodes: n

and then searching the equations to be fulfilled by the amplitudes A , of the modes X,, in order to extend the linear dynamics simply governed by An = snAn to the nonlinear range. From the qualitative arguments developed above, we are led to think that the unstable and neutral modes play the most active role and that the stable modes more passively evolve under the forcing action of the others. Let us incorporate this feature in the formalism itself and assume that the problem is initially posed in the general form:

v = cv +nr(V),

(4.2)

where C accounts for the linear interactions and where n/(V)is at least

4. Nonlinear Dynamics:

from Simple to Complex

119

quadratic in V. An appropriate projection of the equations onto the eigenbasis yields a dynamical system for the amplitudes A, introduced through (4.1), which formally reads:

A n = sn An

+ C gn,mpAmAp +

*

*

(4.3)

mp

where the coefficient Q , , ~ , describes how the interaction of modes X m and X, contributes to the dynamics of mode X,. In fact this projection procedure is rarely achievable without approximation (and further modeling of the physical situation). A favorable case is when the analytical expression of eigenmodes can be handled by hand (or using formal algebra software), for example trigonometric functions as illustrated in Exercises 4.6.1 and 4.6.2. Otherwise one can attempt to extend to the nonlinear domain the Galerkin method already used at the linear stage, e.g. in the second part of Exercise 3.3.2, p. 107. Progress made by replacing the primitive problem expressed in terms of partial differential equations by (4.3), i.e. a differential system of infinite order, may seem slight. However, it is very powerful when combined with the idea of separating driving modes and enslaved modes t o truncate the system, eliminate the latter and obtain an effective system involving only the former. This is the adiabatic elimination procedure advanced as the founding principle of synergetics, the science of nonlinear cooperative phenomena in out-of-equilibrium systems [Haken (1983)l. Let us consider a system on the verge of bifurcating or having just bifurcated ( R x R,). This comes to the assumption that, among all linear eigenmodes, we can isolate a small subgroup of those that are “dangerous,” nearly neutral, either unstable or stable but slightly, i.e. with the real parts u of their growth rates either positive or negative but small. Strictly speaking, the center subspace, subscript ‘c’, is the space spanned by eigenmodes having eigenvalues with u = 0 exactly, and we assume that we have n, such modes. On general grounds, having n, strictly critical modes is a problem of codimension n,, since n, conditions u,(r) = 0 have t o be fulfilled, r being the set of control parameters. In practice, we can vary only one control parameter while remaining in the vicinity of a manifold of codimension n, in parameter space, hence u, x 0 for n, driving modes that by extension will continue to be labelled with the same subscript. On the contrary, all other modes are supposed to remain stable, hence with subscript‘ ‘s’ and a related assumption us < 0, meaning more precisely lSo, ‘c’ may equally well mean ‘central’ or ‘critical’ and ‘s’ either ‘stable or ‘slave’!

120


Fig. 4.3 Left: Spectrum of a linear operator with a group of center modes separated from stable modes by a wide ‘spectral gap’. Right: Adiabatic elimination of stable modes. The whole sets of center-mode and stable-mode amplitudes are featured by axes A, and A,, respectively. A trajectory starting anywhere in phase space rapidly migrates towards a center manifoZd with equation (4.6) which is the nonlinear extrapolation of the center subspace at the origin (A, = 0), then slowly evolves along it according to the effective dynamics (4.7).

that Jo,J 1. Writing X as X = pexp(2~ia:), leads to a clear separation of the modulus p from the phase 527ra of the eigenvalues. The quantity a specifies the angular frequency of the mode that makes the cycle possibly unstable: a: = W / W O . This can be irrational or rational like in the case of periodic forcing. Cases when X E B are special. Equation (4.30) then turns into a real iteration: xk+l

=X

x k

j

(4.31)

and we must distinguish the dynamics corresponding to X > 0 ( i e . a: = 0) for which iterates tend to or depart from the origin in a monotonic fashion from that observed when X < 0 (2.e. (Y = 1/2) for which they evolve by jumping alternatively from one side to the other. Eigenvalue X can be complex or not, but the corresponding stability condition remains 1x1 < 1 and a bifurcation takes place when at least one

142


(pair of) eigenvalue(s) leaves the interior of the unit disc as the control parameters are varied. Different possibilities are indicated by arrows in Figure 4.10 (top-right). Though the main purpose of the course is not a mathematical study of the different scenarios of transition to chaos in the perspective opened by Ruelle and Takens (see Figure 3.5, p. 93), let us mention that one can account for the simplest cases by completing (4.30) or (4.31) with appropriate terms. In the original Ruelle-Takens scenario, the destabilization of the limit cycle still leads to a regular but two-periodic regime (A E C ) . The iteration that accounts for it reads

Z,+l = xz,

- ylZ,l2Z,

,

y = y'

+ iy" .

(4.32)

After the damping of transients, it yields iterates evolving around a circle with radius Jp=-r,where p = 1x1, Fig. 4.11 (top). Trajectories in phase space wound on an invariant limit torus which is the Cartesian product of two circles, one corresponding to the unstable limit cycle (longitude) and

Fig. 4.11 Hopf bifurcation for a map (also called Neimark-Sacker bifurcation). Top: Iteration (4.32) for 1x1 < 1 (left) and 1x1 > 1 (right). Bottom: Perspective sketch of the skeleton of the torus over which the two-periodic dynamics develops asymptotically in time beyond bifurcation.

4. Nonlinear Dynamics: from Simple

t o Complex

143

one for the new mode (meridian section), Fig. 4.11 (bottom), correspond t o this description on the PoincarC surface. The picture obtained in this case is very similar to that for the forced van der Pol oscillator. The difference is just that the two anguIar frequencies in the system now have intrinsic origin, the first one, which plays the role of the forcing, is that of the original cycle and the second one is that of its most dangerous instability mode. They can easily be identified on the records of physical observables, e.g. the two-periodic regime observed in convection and illustrated in Figure 3.8. As before, two-periodicity is observed only in the absence of resonant interaction between the two angular frequencies, that is to say as long as the ratio a remains irrational. Nonlinear couplings are therefore expected t o be responsible for lockings when a approaches a rational value p / q as the control parameters are varied, lockings all the more visible when the denominator q is a small integer. Two-periodic states, and a fortiori locked one-periodic states, are regular asymptotic regimes. In order to observe a chaotic behavior, it seems natural t o think of an effective dimension still increased by one. This was precisely the context of the Ruelle and Takens approach, introducing strange attractors with properties accounting for irregularity in spite of determinism. More precisely, these authors showed (Note 8, p. 92) that when the system is smooth (C”), chaos is generic after supplementary bifurcations introducing two new angular frequencies in the system. The analyticity condition was later made milder ( i e . C2 instead of C”) with chaos possibly occurring in three-frequency quasi-periodic systems.ll However, concrete scenarios from two-periodic regimes to chaos generally rely on different explanatory models which would lead us on mathematical grounds too far away from our main purpose. On the other hand, it is more interesting to spend some time to consider resonant cases. When a = p / q the two periods are resonant. Little is changed if q is large, i.e. q 2 5. The resonance is weak) and the system behaves roughly as in the non-resonant case: typically, the dynamics still takes place on the torus that has emerged from the bifurcation. Genuinely two-periodic unlocked regimes alternate with locked regimes inside narrow tongues as the control parameters are varied. In a quasi-periodic state the trajectory winds on the torus without closing (the torus is densely covered with it), whereas in case of locking the trajectory is a closed cycle wound on the torus, making p turns along the meridian and q turns in the longitudinal l l S . Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, m 5 3,” Commun. Math. Phys 64 (1978) 35.

144

Instabilities, Chaos and f i r b d e n c e

Fig. 4.12 Trajectories on the torus are represented by their phases: Longitude 0 = wot do)(horizontal axis) and meridian coordinate ’p = wt ’p(O) (vertical axis) defined modulo 27r. Up to a factor 27r, the torus is isomorphic to the square 0 E [O; 11, ’p E [O; 11 with opposite sides identified. Left: For a two-periodic state, a @ the whole (0,‘p) square is covered by any given trajectory as t + 00, here (I = (32/71)’12 = 0.671345.. . . Right: When a = p / q , the same state is reached after a time qT0 = p T , here a = 2/3 = 0.666666.. . . (Notice that a = 2/3 is in principle a case of strong resonance; this value has been chosen here only because the figure is easy to read.)

+

+

0,

direction (Fig. 4.12). Chaos may enter after a new instability mode sets in. By contrast, when q 5 4 the resonance is strong and has marked effects, destroying the torus over which the motion takes place. Most extreme cases correspond to q = 2 and q = 1 that bring back the map from a complex iteration (4.30) to a real iteration (4.31) since for p / q = 1 / 2 one has X = exp(i7r) = -1 and for p / q = 1/1, X = exp(2i7r) = 1. Each of these two cases govern a specific scenario. 0 The subharmonic cascade observed in convection (Fig. 3.7) corresponds to the strong resonance a = 1/2, whose normal form reads: Xk+l

= fr

(Xk)

= -(I

+

- xi.

T)xk

(4.33)

The corresponding scenario is illustrated here using the so-called logistic map12 Xk+l

= 4 r x k (1 - x

k )

7

(4.34)

with 0 5 r 5 1. The full bifurcation diagram in Figure 4.13 displays how the attractor (values of X reached as the iteration proceeds, after elimination 12For an introduction, see R.M. May, “Simple Mathematical models with very complicated dynamics,” Nature 261 (1976) 459-467.

4.

Nonlinear Dynamics: from Simple t o Complex

Fig. 4.13

145

Subharmonic cascade for (4.34)

of transients) changes as the control parameter T is varied. For T < 314, a single point is obtained, corresponding t o a periodic trajectory with period one (period TOif one considers the underlying limit cycle). At T = T I = 314 a first period-doubling takes place: a period-two regime sets in with two values XI et X, alternatively visited as the system evolves. (Working out the nonlinear variable change that turns (4.34) into the normal form (4.33) close to the bifurcation point is an interesting exercise.) At T = r 2 = 0.86237.. . , this period-two attractor is destabilized against a period-four regime, etc., as observed in convection experiments (Fig. 3.7, p. 95). The bifurcation cascade ends up at T = T , = 0.89248. . . , where the system becomes periodic with infinite period in the form 22n, n -+ 00. At this value, the so-called margin of chaos, the system is in fact aperiodic since it never returns exactly in the same state. Beyond T,, continuous vertical segments, in fact densely covered with points, are observed on the bifurcation diagram for each value of T . They are the trace of chaotic attractors for which infinite sets of unequally distributed points are visited all along a trajectory. In fact, the situation is also very complicated in the parameter space since the return t o periodic attractors is clearly visible in some parameter windows, the corresponding periodic states themselves decaying to chaos


146

through subharmonic cascades. Programming (4.34) or its variant (4.69), p. 175, is so easy that one should not miss exploring their bifurcation diagram numerically by oneself. 0 Interrnittency (type I, that featured in Figure 3.9, because several other types exist) takes place at a l/l-resonance generically accounted for by the map:

Before the transition, as long as T

x*=r+x*+x;

< 0, it has two fixed points =2

solutions of:

X*=fJ-r.

All trajectories starting with X < X ( + ) (and small enough) converge towards the fixed point at X ( - ) . Other trajectories leave this limited region of phase space, Figure 4.14. Beyond the transition, for T > 0, the fixed points have disappeared but the local structure of phase space keeps a track of their presence as “ghosts”: iterates go through the kind of tunnel opened between the graph of the map and the line Xk+l = x k . They move very slowly since X k f l = XI, plus some tiny correction, approach the origin and then go away. Now the global structure of phase space may provide the opportunity for trajectories visiting remote regions of phase space to come in the neigh-

t

Fig. 4.14 Type-I intermittency according to (4.35). Left: r < 0, two fixed points, X ( - ) stable, X ( + ) stable. Right: r > 0, no fixed points but a narrow tunnel between the graph of the map and the line Xn+l = X,.

4 . Nonlinear Dynamics:

from Sample to Complex

147

borhood of the origin. When this is the case, for T < 0 the system coming close to the stable fixed point eventually converge to it. By contrast for T > 0 fixed points no longer exist, iterates sent in the vicinity of the origin, travel slowly through the tunnel, then get away and come back later, hence the intermittent behavior observed in Figure 3.9, p. 97. Laminar intermissions are uncorrelated and their durations are unpredictable. This is due to the more or less uncontrolled evolution far from the previously existing limit cycle which generates randomness in the reinjection process. While interpreting the experiments, one must remember that the map describes what happens after an appropriate section of the dynamics has been performed. At the bifurcation, twin limit cycles, one stable and the other unstable, collide and disappear. Laminar intermissions are made of slowly evolving regular periodic oscillations corresponding to the “ghost” limit cycles. They are interrupted by chaotic bursts during which the system explores remote regions in phase space, until it is sent back in the laminar region, ready for a new intermission.

4.3

Characterization of Chaotic Regimes

As conceived by Ruelle, chaos is a dynamical regime characterized by a specific sensitivity of trajectories to initial conditions and small perturbations. In phase space, it is accounted for by strange attractors that are robust attracting limit sets on which small departures between two trajectories are indefinitely amplified (the general meaning of instability) as a result of a stretching-and-folding process analogous to the making of Danish pastry (baker map, see later Figure 4.17). The main interest of these concepts is to reconcile determinism and stochasticity in dissipative dynamical systems, which was initially supposed to further shed some light on the problem of the “nature of turbulence’’ as discussed in Chapter 3, $3.2.2.3, p. 92. In this section we thus focus on a quantitative estimate of the two main facets of chaotic dynamics: the “longitudinal” instability measured by L yapunov exponents and the fractal dimensions characterizing the %ansverse” foliated structure.

148

4.3.1


Instability of trajectories and Lyapunov exponents

The instability of trajectories on a strange attractor is illustrated here by means of an everywhere expanding iteration called the dyadic m a p (Fig. 4.15, top-left). It reads:

Xk+l = 2Xk

(mod 1).

(4.36)

It is convenient to visualize the trajectory {Xk, k = 0 , 1 , . . .} as a walk in the complex plane Zk+1 = Zk exp(27riXk). ( 2 has no dynamical significance here by contrast to X which is governed by the map.) This simple

+

Fig. 4.15 Interpretation of chaos. Top left: Dyadic map. Top right: Divergence of neighboring trajectories. Bottom left: Dyadic walk. Bottom right: Random walk.

4 . Nonlinear Dynamics: from Sample to Complex

149

representation helps us to clearly illustrate the divergence of trajectories starting at neighboring points (Fig. 4.15, top-right) and the long-term evolution (Fig. 4.15, bottom-left) that leads one t o think of the random walk that would be obtained by drawing the successive Xk uniformly at random over the unit interval (Fig. 4.15, bottom-right). The divergence rate of trajectories is a good measure of chaotic behavior. Let us show how the analysis proceeds in a simple example and consider a map f of a single real variable X :

Xk+l = f ( x k ) 7 more general than (4.36). To begin with, a given trajectory { x k ;k = 0 , 1 , 2 . . .} starting at X O = X(O) is taken as a reference and we consider a neighboring trajectory { X k ;k = 0 , 1 , 2 . . . } starting at XO = X(O) 6 x 0 . Denoting f' = d f /dX we have:

+

X l = x1

*

+

ax1

a x 1

=f(X0

+ 6x0)= f ( X 0 )+ f ' ( X 0 )

a x 0

= f'(X0)6Xo

and using the chain rule:

Assuming a geometrical growth/decay as l d x k l which defines both y and X = lny, we get:

N

yk1dXol

= exp(kX)lbXo(,

At this stage, y still depends on k. To get rid of this dependence we take the limit k

-+ 00

which leads to the definition of the Lyapunov exponent:

(4.37) In other words, the Lyapunov exponent is thus the time average of the local divergence rate In(lf'l), i.e. a measure of long term unpredictability. Trajectories diverge in the mean when X > 0, which can be taken as the definition of a chaotic dynamics. For the dyadic map (4.36), one gets X = In 2 > 0.

150


The extension to maps with several variables leads t o the definition of the Lyapunov spectrum, which rests on the analysis of the asymptotic behavior of the product of Jacobian matrices obtained from the chain rule. This generalizes the eigenspectrum of the stability matrix at a fixed point t o the case of arbitrary trajectories. Technically, the matter is difficult but, in the limited context of this course, it is sufficient to know that this can be done and that, by successive generalizations, one can determine Lyapunov spectra for differential systems (from time-7 maps, see p. 28) and for continuous media governed by partial differential equations (after spectral approximation or discretization leading to finite-order ordinary differential systems, see Appendix B). The Lyapunov spectrum can be ordered by decreasing values of the individual exponents. The system is then declared t o be chaotic when the largest exponent is positive. The empirical determination of Lyapunov exponents from time records will be examined later, p. 165. Let us stress the fact that the whole approach in terms of dynamical systems is a progressive extension of linear instability concepts from timeindependent regimes (fixed points of continuous-time systems) to periodic regimes (limit cycles then seen as fixed points of discrete-time systems) and finally to irregular aperiodic regimes. 4.3.2

FTactal aspects

Up to now we have been interested in expansion properties “along” the attractor. However the systems of interest are supposed to be dissipative, which implies overall contraction of volumes in phase space. As we have seen, chaotic behavior is marked by instability, which means expansion in some directions. This expansion has to be more than compensated for by stronger contraction in other directions. Some folding must then take place in order t o maintain trajectories in a bounded region of the phase space, as sketched in Figure 4.16. (The case of regular regimes would be much less anisotropic with, at most, neutral directions on average.) What happens in the “transverse directions” can be concretely illustrated using a celebrated simple two-dimensional map called the baker map (Fig. 4.17, top-left). This map of the unit square [0,1] x [0,1] expands it onto itself by a factor n = 2 in the direction of the first coordinate and by a factor n’ = a/. in the other direction (6’< 1 j contraction). When a = 1, that is n’ = 1/2, the baker’s map is conservative (areas are preserved). In order to obtain a dissipative system, contraction must

4 . Nonlinear Dynamics: from Simple

dissipation + instability

to

Complez

151

nonlinearity

Fig. 4.16 Expansion in unstable direction, contraction due to dissipation, folding by nonlinearities, all in combination, produce horseshoes typical of chaotic attractors.

be larger than expansion, i.e. a < 1, so that KK' = a < 1. The strange attractor obtained in that case displays a characteristic fractal transverse structure (Fig. 4.17, top-right). This fractal structure is a triadic Cantor set classically obtained by removing the middle third of a segment and repeating indefinitely the operation on the two segments left apart in that operation as shown in Figure 4.17 (bottom), yielding a self-similar set that is invariant upon magnification by a factor of three.

xz

0 X,

Fig. 4.17 Top-left: Conservative baker map (1st step: expand along X1 ( K = 2) and contract along X z (K' = 1/2); 2nd step: cut right half of rectangle and place it back in square). Top-right: Strange attractor for the dissipative map with K' = 1/3 (expansion rate KK' = 2/3 < 1). Bottom: Few steps of the construction rule of the triadic Cantor set, to be read from right to left. (Notice that, to build the dissipative baker attractor, we used a map that translates part (2) above part (1) and puts it upside down so that the fractal structure follows from the conventional Cantor middle-third rule.)


152

15

1

07

Y 06

05

- 1 51

-15

-10

-05

00

05

X

10

I

15

05 O6

x

or

Fig. 4.18 Henon attractor obtained by iterating map (4.38, 4.39) for a = 1.4 and b = 0.3 from some initial condition after elimination of the transient. Left: The full attractor. Right: Zoom on the boxed region.

In practice, nonlinearities usually fold the trajectories as seen with the HCnon map: l 3 (4.38) (4.39) The attractor corresponding to a = 1.4 and b = 0.3 is presented in Figure 4.18. The folding originates from the nonlinearity in (4.38), which is of “logistic” type (see Exercise 4.6.9). Coefficient b in (4.39) guarantees the dissipative character of the map provided that Ibl < 1, as shown by performing Exercise 2.5.1 with this map. The fractal character of the attractor is particularly obvious from the magnification of the boxed region displayed in the right part of the figure. The fractal dimension gives a good idea of the way the attractor occupies the phase space. In a d-dimensional space, it is obtained by covering the considered set, here the attractor, by elementary volume elements of size &, next counting the number N(E)of such elements necessary to cover it, and finally study how that number grows as the linear size E of the volume elements tends to zero. The fractal dimension df is thus defined through 13M. HBnon, “A two-dimensional mapping with a strange attractor,” Commun. math. Phys. 50 (1976) 69.

4.

Nonlinear Dynamics: from Simple to Complex

153

the formula: (4.40)

For an ordinary, connected, continuous set, this definition yields the usual topological dimension (0 for a point, 1 for a line, 2 for an ordinary surface, e t c . ) . Let us see how it works for an indefinitely fragmented object like the triadic Cantor set by considering it at step k of the construction process. It can be covered by segments of length E = 1/3k, and 2k such segments are needed at that step, hence:

somewhere between the dimension of a countable set of isolated points (0) and that of a continuous segment (1). Here the evaluation does not depend on k because the set is strictly self-similar but more generally the limit E + 0 must really be taken. The computation of the fractal dimensions of other classical self-similar sets from a direct application of (4.40) is the subject of exercise 4.6.12. Turning to the dissipative baker map, let us focus on what happens in the transverse direction Xz. We take for granted that, owing to the expanding character of the map along XI,the attractor is continuous in that direction, which just adds 1 to the dimension found for the transverse part. Following the same idea as for the triadic Cantor set, we can observe that, after one iteration, the length of a segment along X Z is multiplied by K‘ < 1, which suggests us to take E = ( K ’ ) ~after k iterations. Each application of the map brings about K reduced copies of the full set, so that we have N ( E = ) K ~ From . the formula, we get df = log(K)/log(l/K’) along the contracting direction and thus 1 + log(K)/log(l/d) for the full attractor displayed in Figure 4.17 (middle part). With K = 2 and K’ = a / & with a = 213, this yields df M 1.63. As an exercise one can look at the aspect of dissipative generalized baker maps with definitions more complex than that illustrated in Figure 4.17, e.g. with K > 2 and different values of a for different pieces, and then try t o determine the dimension of the corresponding (transverse) Cantor sets, which might be less simple than it seems.

154

4.4

Instabilities, Chaos and Tvrbdence

Empirical Approach of Chaotic Systems

Let us consider a chaotic regime observed in a given experiment, e.g. in convection. The need of an empirical approach becomes obvious when it appears impossible from a practical point of view to get an ab intio understanding of its nature. Most often, even something as qualitative as the type of transition scenario that develops under specific conditions cannot be predicted. For example, in Chapter 3, $3.2.3,we have seen that not only the physical properties of the fluid matters, but the experimental configuration and, in case of attractor coexistence, the history of the experiment also play a role in the transition (though the system, when engaged in a given scenario, follows all its steps at a quantitative level). Sufficiently far from the threshold of the primary instability, a multiplicity of different possible permanent regimes can be reached by following specific experimental procedures. The effective phase space is thus already very complicated and poorly understood, even when confinement effects select a small number of driving modes. It is therefore fully justified to spend some time to seek a representation of the dynamics in some reconstruction space obtained from the experimental records, a space in which the evolution can be described, the amount of chaos can be measured and, hopefully, techniques of control can be developed. The output of experiments, either in the lab or using a computer, generally presents itself in the form of time series of some observable, i e . a series of numerical values taken by some function W = W(X) of the system’s state X = ( X I , .. . ,X d ) in its phase space X with dimension d.I4 Let us assume a regularly sampled time series: {Wk,k

= 0,1,. . .}, with Wr, = W ( X ( t k ) ),

tk

= kr ,

where T is the inverse of the sampling frequency. Just having a look at the plotted time series is a step that should never be skipped since this gives one valuable information on the recurrent or intermittent character of the signal. But clearly more objective analyses are required, especially when the system has reached some sort of attractor, i.e. a permanent regime characterized by statistically stationary signals, with distributions that do not change significantly as times goes on, and for 14For systems evolving in space, the output may also be pictures. This opens the vast field of image processing, basically extending to two-dimensional quasi-continuous arrays the viewpoint of one-dimensional, discrete scalar series on which we focus here exclusively.

4 . Nonlinear Dynamics: from Simple

t o Complex

155

which finite-length series are typical of the dynamics, provided that they are not too short (i.e. correspond to several turns around the attractor). The first operations to perform on a time series of length K , {Wo,W1,. . . ,W K - ~ }are , of course to compute its average:

and its variance:15 K-l gz

--

W - K

k=O

and then t o rescale it by making the changes:

For simplicity, we assume in the following that this preprocessing has been performed, i.e. that we work with a signal supposed to be stationary, with zero mean and unit variance. In a second instance, one usually considers the autocorrelation of the signal defined as:

C ( A t )= lim

T+m

1 T

T W ( t ) W ( +t A t ) d t

(4.41)

In the case of a finite series of discrete records, we get:

1 K-n-l C(K) = K - K k=O

WkWkfn,

(4.42)

but when the series is very long and as long as the shifts IC are small enough (IC < < K ) , any difference with the result of the discrete evaluation of the integral (4.41) for At = KT is not perceptible. The autocorrelation is the simplest device to identify periodicities in the signal. It is usually not computed from its definition but rather through Fourier transforms. The next section thus begins with a brief reminder about them and continues by introducing the Hilbert transform, a useful tool t o perform the demodulation of periodic signals with superimposed slow amplitude and phase variations. 15The “unbiased” variance should be defined by dividing with K - 1 instead of K but we are always interested in K large so that this makes no difference.

156

4.4.1


Standard analysis b y means of Fourier transform

On general grounds the direct Fourier transform is defined by:

1

F+00

#.

f(w)= 2x

f ( t ) exp(-iwt) dt

'

-00

and the inverse transform by:

f(t)=

J(w) exp(iwt) dw . --co

The Fourier transform of the autocorrelation function then reads: S(w) =

2x

/

00

-m

C ( T )exp(-iw.r) d7

and a straightforward computation shows that

where W ( w ) is the Fourier transform of W(t). S ( w ) is called the Fourier spectrum of the signal.16 Direct inspection of the signal sometimes suggests that one has to deal with a periodic process that is modulated in amplitude and/or phase. An elegant way to perform the demodulation consists in constructing a complex signal Z ( t ) , whose real part is the primitive signal W ( t )and whose imaginary part is constructed so as to make a x/2 phase angle with it. This is easily understood from the consideration of a strictly periodic signal: let W ( t )= Acos(wt), the signal at 7r/2 is then Acos(wt-x/2) = Asin(wt), so that Z ( t ) = Aexp(icp(t)), with amplitude A = 121 and phase cp, such that dcpldt = w is the angular frequency. In the general case, the instantaneous amplitude of the modulated signal is then given by the modulus of Z and its instantaneous period is derived from its argument by differentiating it with respect to time, provided that one knows how to obtain the signal at x/2. This is done from the Fourier transform of W ( t ) ,by copying the case of the periodic signal taken as an example above, frequency by frequency. Starting with:

[,

+m

W ( t )=

W ( w ) exp(iwt)dw ,

(4.43)

leThis result is called the Wiener-Kintchine theorem. The derivation uses a few tricks, among which the fact that the Fourier transform of a constant f 1 is a Dirac distribution 6 ~ ( w ) .

4. Nonlinear Dynamics:

from Simple

to CompIex

157

and setting:" 1

* ( w ) = ;[A(w) L

+ i B ( w ) ]= JV(-w)*,

(4.44)

one can observe that A ( w ) and B(w) are even and odd functions of w, respectively. In full generality one gets B(0) = 0 and, further also A(0) = 0, since W ( t )is assumed to have zero mean. Expression (4.43) can then be rewritten as: +W

[ A ( w )cos(wt) - B ( w )sin(wt)] dw,

W ( t )=

which allows one t o get the imaginary part, here denoted W ' :

1

+03

W ' ( t )=

[ A ( w )cos(wt - n/2) - B(w) sin(wt - n/2)]dw,

which more simply reads:

W ' ( t )= l + m [ B ( wcos(wt) ) + A ( w )sin(wt)]dw. By definition of Z = W

+ iW',one then obtains:

1

+m

Z(t)=

exp(iwt)[A(w)

+ iB(w)]dw,

(4.45)

which involves a sum over the positive angular frequencies (the so-called 'analytical signal'). Comparing (4.45) and (4.43, 4.44), one obtains the Fourier transform Z ( w ) by setting to zero all components of the Fourier transform of W corresponding to negative angular frequencies and by doubling all the others. The analytic signal Z ( t ) itself is then recovered by computing the inverse transform of 2. Mathematically W and W' are Hilbert transform of each other" hence the expression 'Hilbert transform demodulation' (the full procedure is implemented by hilbert .m in the MATLABsoftware). 4.4.2

Reconstruction by the method of delays

Let us now take a point of view more in line with the theory of dynamical systems. A difficulty arises immediately from the fact no specific assumption can be made, except that a deterministic framework is relevant, so that 17The complex conjugation property arises from the fact that the observable W is supposed to be a real function of X . 18Formally, I?l'(w) = -(l/n) P l_",[I?l(w')/(w - w')]dw', where '7 denote the Cauchy principal part of the integral.

158


one can just write formally X = F ( X ) for unspecified states living in some space X. In particular, the nature and the dimension deR of the manifold supporting the dynamics,lg and the explicit relation between states and the observable W = W ( X ) are not known a przorz. Practically all reconstruction techniques derive from the method of delays mathematically formalized by Takens in 1981,” which is numerically more robust than a previous approach based on the evaluation of successive time derivatives of the experimental signal in terms of finite differences.21 Let us illustrate the reconstruction approach using a discrete-time system for simplicity. At the beginning we assume that

Xk+l = F ( X k ) and that the time series of some scalar observable W is available (the method is, at least conceptually, easy to extend to the case of several observables). Reconstructing the dynamics means determining an empirical relation between the xk in their phase space only from the knowledge of the W,, lc = 0,1,. . . A single measure WO = W ( X 0 ) is not sufficient t o determine the state X O since we surely need more than one coordinate t o define it. But we assume that the next value W1 corresponds to a point X 1 that evolves from X O under the map 3,unknown but existing. The second measurement thus adds a piece of information about the coordinates of X O through W1 = W(X1) = W ( F ( X 0 ) ) .The third one, WZ = W ( X 2 ) = W ( F ( X 1 ) ) = W ( F ( F ( X 0 ) ) )adds another piece, etc. In principle, a sufficiently long series of dtest successive measurements, { W o ,. . . ,Wdtcst -l ), should serve us to specify X O . In the same way, {Wl,. . . , W d t e s t }would do so for X I , etc. Eventually a whole trajectory would then be reconstructed from the series of vectors in ~ ~: ~ e a ~

The concrete implementation of the method is by increasing the dimension of the reconstruction space until a consistent quantitative assessment of the observations can be given. In practice, the problem can be reformul9The dimension of the physical system itself is usually infinite; think of continuous media. z°F. Takens, “Detecting strange attractors in turbulence,” Lect. Notes Math. 898 (1981) 366-381. 21N.H. Packard, J.P. Crutchfield, J.P., J.D. Farmer, R.S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45 (1980) 712-716.

4.

159

Nonlinear Dynamics: from Simple t o Complex

lated in terms of the reliability of the reconstruction. The space of vectors v k has to be in correspondence with the region of phase space visited by the system when the permanent regime is reached. Whereas we can accept redundancy, i. e. a dimension dtest that is too large, we must not lose useful dynamical information. In mathematical terms the representation of the system must be injective, so that different states have different reconstructions:

What was defined as a tentative number of component dtest is thus more mathematically understood as the dimension of the space in which the effective phase space can be embedded by means of some injective map. To stick with this abstract point of view more closely, we now define dtest as the embedding dimension, denote it d,, and thus specify states in the embedding space as: v k

= [wk;. . . ;w k + d , - l ]

(4.47)

7

Takens’ method of delays is sketched in Figure 4.19. His theorem states that the VI, defined above, where the observable W is defined by a differentiable functional on phase space W , achieve a reliable reconstruction provided that the dimension de is large enough: d, 2 2d,ff+ 1, where d,R is the topological dimension of the manifold supporting the effective dynamics. This theorem can be understood as an extension to the present context of Witney’s theorem that states that a compact d-dimensional differentiable manifold M can be embedded in an Euclidean space As an illus-

t tk

x

Fig. 4.19 Illustration of Takens’ theorem. M is the manifold in the space over which the effective dynamics takes place. Coordinates of points representing the system in Rde are obtained from the series of measurements wk through (4.47).

160

Instabilities, Chaos and ‘hrbdence

tration of the latter, one can consider a loop (dimension 1) that would be projected on a plane as the figure eight. In order t o resolve its structure, and in particular to check that the intersection is a fake, one must be able to look a t the loop from another side, i.e. t o stand in a 2 x 1 1 = 3 dimensional space. In fact the reconstruction proposed by Takens is more general than that proposed by (4.47) since the v k can be any series of d, measurements, [Wk;W k + n l ; ...; Wk+nd,-l], and nothing forbids it to take irregularly distributed intervals K ~ .It is however natural to take K~ as the successive multiples of some basic K , i.e. K~ = Q K , Q = 1, 2,.. . , d, - 1. When the signal is obtained from the time sampling of a continuous-time system with period T this corresponds to a sub-sampling at period KT. The mathematical viewpoint developed so far is apparently strong. However our enthusiasm must be somewhat tempered and a pragmatic perspective has to be taken, for it is not clear that the physical systems we are interested in fulfill the theoretical conditions underlying the theorem, and first of all that the d,R-dimensional manifold over which the dynamics takes place is sufficiently smooth. As it is usually the case for chaotic nonlinear systems, the attractor has a fractal dimension df 2 d,a and, following Maiie (1981), one can replace d,R by df in the inequality for d,. This still does not lead to any concrete estimate, in part because measurements are always polluted by “noise” that comes and hinders the reconstruction. Strategies have thus been developed to get around these difficulties and determine more or less optimally the two basic ingredients of any reconstruction: the base delay K and the embedding dimension d,. This is what we now briefly discuss, inviting the interested reader t o consult, e.g. [Abarbanel (1996); Kanz and Schreiber (1997); Weigend and Gershenfeld (1993)] for more detailed approaches.

+

4.4.3

Sampling frequency and embedding dimension

For more specificity we use a synthetic signal obtained by numerical simulation of a noisy limit cycle governed in the complex plane Z = X + ZY by a Langevin equation

2 = (1 + i ) Z - (1 - i)lZ12Z + [ ( t ) , where [ ( t ) is a Gaussian noise. We assume that the trajectories are computed by a second order Runge-Kutta scheme (B.9-B.lO) with time step 6 t = 0.01. A noise with amplitude 0.05 is added to X and Y at each time

1. Nonlinear Dynamics: from Simple to Cornplez

161

Fig. 4.20 Top: Original noisy periodic signal in its own phase space. Bottom: Reconstruction with sampling time T = 0.03 and delays K. = 5, 25 and 45 of observable

=

W(t) X(t).

step (for the generation of Gaussian noise see [Press et al. (1986)l). The trajectory used for this example is displayed in Figure 4.20 (top). The period of the (deterministic) signal is T = r. The signal W ( t )that we take is just the X component of the trajectory and we mimic the sampling process by recording its value every T = 3dt. The first problem is that of the sampling time, and thus that of K . An exaggerately high frequency is not an advantage since a huge volume of data is produced, the most part of which is redundant owing to the noise that blurs the information. It is then believed that the delay between two successive records must be sufficient to bring novel information. A practical rule is t o take the delay corresponding t o the first zero of the autocorrelation as the effective sampling time M-. For the present (deterministic) sinusoidal signal, W ( t )= sin(2t) , the autocorrelation defined through (4.41) is C ( A t ) = cos(2At) and this rule prescribes A t = r / 4 , i.e. a quarter of a period: two measurements out of phase by such a shift are indeed fully discriminated since one is maximum when the other is zero. The discrete estimate of the autocorrelation function (4.42) is displayed in the left part of Figure 4.21 where one can observe that the noise adds

162


-0 8 0

K 50

100

150

m

0

Fig. 4.21 Autocorrelation C ( K )and mutual information Imut(n) of signal W in Figure 4.20 as a function of the delay n.

a slight damping to the behavior expected for a periodic signal. Applying the rule leads t o K N 25, which indeed corresponds t o a quarter of a period once recalled that T = 36t = 0.03. Two-dimensional reconstructions in the plane (Wk,Wk+n)for K = 5, 25 and 45 are displayed in Figure 4.20 (bottom row) which shows that the representation with K = 25 is the most similar to the original signal, whereas for K = 5 and K = 45, corresponding to measurements too close in time or practically in phase opposition, the reconstructions align along directions Wk+n = -kwk and Wk+n = -Wk, respectively. The study of the autocorrelation function, of linear essence, does not always lead t o a satisfactory choice of K . Let us now mention a more general criterion resting on a similar philosophy but with a better nonlinear flavor. This criterion bears on the mutual information contained in two records shifted by some time amount when compared to that contained in a single record. Again, stationarity of the signal is assumed. First the probability distribution P ( W ) of wk = W is obtained by an appropriate normalization of the histogram of the values of wk. Next, the joint probability Pn(W',W " ) is determined in the same way but for couples (wk = Wr;Wk+&= W " ) , with K = 1 , 2,.... From these, the mutual information is defined as:

This quantity is a measure of the redundancy in the signal: when K is small, points in phase space are highly correlated so that learning about

4 . Nonlinear Dynamics: from Simple to Complez

163

Wk+n when wk is known does not bring much novel information; by contrast when n >> 1, the points become uncorrelated and P n ( w k , w k + n ) is essentially p(Wk)p(Wk+n),so that the mutual information is nearly zero, the information gained is the same as from independent drawings wk and W k + n using P ( W ) . As can be seen from the right part of Figure 4.21 that displays the graph of Imut(n)obtained by applying (4.48) to our signal, the mutual information does not decrease monotonically but in general presents a first minimum at some intermediate K that defines an optimal value nopt corresponding to a minimum of the redundancy, before ultimately decaying owing to statistical decorrelation: taking simultaneously Wk and W k + n O p t should thus give the best information about the dynamical evolution. Here, without surprise nopt is the same as that given by the previous rule, but more generally this is not the case. It should be noted that, when no clear minimum of Imut(n)is obtained, this can mean either the presence of a very large noise, or that the observable has been under-sampled, or that too many degrees of freedom are involved, all cases where methods of the theory of low dimensional deterministic dynamical systems are of little help. Once the parameter IE has been determined optimally, one can re-sample the time series with the new step and discard redundant information contained in the intermediate values that are too closely correlated to the retained ones. In fact, the n parallel time series obtained by changing the phase of the reconstruction modulo IE are not independent but yield K equivalent reconstructions of one and the same trajectory. Comparing these reconstructions may serve to appreciate the amount of noise quantitatively. Renumbering the time series, we can assume n = 1 and focus on our second problem: finding the most appropriate embedding dimension d,. Keeping in mind that we want to obtain an injective representation, we reverse the implication (4.46),now stated: Xk=Xk’,

V k = V k ’

which suggests to analyze the reliability of the state identification (r.h.s.) from comparisons of different reconstructions (1.h.s) especially in spaces with different dimensions: neighbors in some space are “true neighbors” if they remain so for all reconstructions. This property can be checked by increasing d,, i.e. enlarging the width of the window dragged on the data. The method of false neighbors is an efficient strategy to decide when to stop adding coordinates. Consider a trial dimension d’, i.e. [wk; wk+l;.. ;W k + d ’ - 1 ] and the (d’ 1)-dimensional reconstruction ob-

.

+

164


Fig. 4.22 Comparison of the reconstruction in a d’-dimensional space featured by the horizontal axis (coordinates { w k ; . . . ; W k + d f - l } , and a reconstruction in the (d’ + 1)dimensional space obtained by adding coordinate W k + d ! .

tained by adding a component W k + d l . Choose a distance in reconstruction space and a criterion to decide which is neighbor and which is not (depending on the noise amplitude). Next determine the number of false neighbors, i.e. the number of pairs of points that were neighbors in d‘ dimensions and are no longer neighbors in d’ 1 dimension, as illustrated in Figure 4.22. Then increase d’ up t o the point when the fraction of false neighbors decreases significantly and choose that value as optimal embedding dimension. The Euclidean distance derived from the L2 norm is usually not a convenient choice for neighborhood evaluations, since it requires a lot of computations. By contrast, the distance derived from the Lm-norm, i . e .

+

I.

dist (V(’),V(’)) = sup IV,(’) - V,(’) k

only requires comparisons and is most economical. Choosing the embedding dimension as given by the false-neighbor method should appropriately unfold the structure of the attractor. The final presentation of the result can be improved by changing from the canonical basis in the reconstruction space t o a basis correlated t o the data in the least-square sense.22 The method known as proper orthogonal decomposi22(a) The idea was introduced by D.S. Broomhead and G.P. King, “Extracting qualitative dynamics from experimental data,” Physica D 20 (1986) 217-236. (b) For a concrete implementation, see R. Vautard, P. Yiou, and M. Ghil, “Singular spectrum analysis: a toolkit for short, noisy chaotic signals,” Physica D 58 (1992) 95-126.

4. Nonlinear Dynamics: f r o m Simple t o Complex

165

“neighboring”trajectoly ortion

v ,. Fig. 4.23 Empirical determination of the largest Lyapunov exponent of a system in its reconstructed phase space. (a) Reconstructed attractor. (b) Rescaling of the distance between the reference portion of trajectory and its neighbor.

t i o n or singular value decomposition is implemented in numerical softwares such as MATLABand is also used in the field of pattern recognition, which may be of interest in the analysis of space-time chaos to be introduced in Chapter 5.

4.4.4

Application

From a d,-dimensional reconstruction one can next extract quantitative information of the system, and in particular the amount of chaos present, using quantities such as Lyapunov exponents or fractal dimensions of the attractor. By contrast with the theoretical approach, in which the expression of the dynamical system is known, here we have just a (very long) time series of some observable at given control parameter. The attractor is then first reconstructed by the method of delays from this time series, Figure 4.23(a), and we must assume at this stage that the permanent regime is reached and that the system explores its attractor repeatedly and satisfactorily in a statistical sense. If this is the case, when the system comes in a given region of phase space, it never comes exactly at the same place but in some neighborhood. Taking a portion of the trajectory as a reference, the vector field in its neighborhood can be estimated from the set of trajectory pieces followed in close approaches to that reference portion, e.g. through a least-square adjustment of the coefficients of the local Jacobian matrix. This is in general a hard matter. A quantity that can however be determined more easily

166


Fig. 4.24 Determination of the correlation dimension in a convection experiment. Left: Distribution of distances in phase space for several embedding dimensions. Right: Variation of the slope at small R as d, is increased. Exponent Y is seen to saturate at 2.8 which is the correlation dimension of the considered attractor. When d, is too small, trajectories fill the reconstruction space homogeneously, as if the signal was a white noise, hence the corresponding line Y = d,. After Malraison et a[., Note 25.

is the largest Lyapunov exponent. The procedure used in early attempts23 reflects particularly clearly its nature as an average trajectory divergence rate: since Lyapunov exponents are quantities relative t o the tangent evolution, the distance between pieces of trajectories serving t o compute the divergence rate must remain small enough; when following the reference trajectory, one is therefore obliged to look for pieces of trajectories as close as possible to the current point, in the direction of fastest divergence, as suggested in Figure 4.23(b). A quantity that can serve to characterize the fractal properties of strange attractors is the correlation dimension introduced by Grassberger and Pro~ a c c i a It . ~is~extracted from the distribution of distances between pairs of points on the reconstructed attractor A(Vi, Vj),obtained by computing 1 C ( R )= lim N+ca N 2

T(R-A(Vi,Vj)),

(4.49)

{Xi,Xj}

where T = 0 for u < 0, and T(u) = 1 for u > 0 (Heaviside distribution). Provided that the statistic is sufficient, this quantity measures the number of points in volume elements of radius R around each point, roughly speaking its mass, as it varies with R. For a compact d-dimensional object, one would have C ( R ) Rd as R + 0. The correlation dimension is thus

-

23A.Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 10 (1985) 285-317. 24P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9 (1983) 189-208.

4. Nonlinear Dynamics: from Simple to Complex

167

defined as (4.50) The way it is defined makes exponent v similar to df. As such, it gives an easily determined measure of the fractal character of the attractor from the statistics of a long time record in the embedding space. This approach has been developed many times. The example given here is the first application to a concrete convection experiment in confined geometry, by Malraison et ~ 3 1 . It~ ~is seen in Figure 4.24 (left) that the correlation integral (4.49) plotted in log-log scale as a function of R indeed display a linear part at small R from which an exponent v can be derived. At the time of the experiment, the determination of the optimal embedding dimension was made by just increasing de progressively. If the embedding dimension is chosen too small, one observes that v is close to d, which is easily understood from the fact that the reconstruction does not contain enough information to evidence the deterministic character of the dynamics. Points fill the V-space homogeneously as a random signal would do. On the contrary, if upon increasing d,, it happens that v saturates at some finite value, as here, it is a good indication that some deterministic dynamics is at work producing a low dimensional chaos, see Figure 4.24 (right). If unfortunately, v continues to increase with de, either confinement effects are too weak and the effective signal is not low dimensional, or the noise level is too high. To conclude this section, let us remark that a reliable quantitative determination of the amount of chaos requires a lot of data but that useful information can anyway be obtained from the reconstruction technique, e.g. to attempt controlling chaos.26 Software packages are available for an automatic treatment of experimental data, e.g. Note 22(b). 25B.Malraison, P. Atten, P. BergB, M. Dubois, “Dimension d’attracteurs Btranges: une dBtermination expkrimentale en regime chaotique de deux systknes convectifs,” C.R. Acad. Sc. Paris 297 SBrie I1 (1983) 209-214. 26Chaos control and its applications have received considerable attention in the 1990s. See, e.g. T. Shinbrot et al., “Using small perturbations to control chaos,” Nature 363 (1993) 411-417; S. Hayes et al., “Communicating with chaos,” Phys. Rev. Lett. 70 (1993) 3031; for another approach, see K . Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170 (1992) 421-428, and W. Just, et al, “Mechanism of time-delayed feedback control,” Phys. Rev. Lett. 78 203-206. A more complete presentation can be found in [Schuster (1999)l.

Instabilities, Chaos and ‘Ihrbdence

168

4.5

Conclusion

As a whole, one can say that the problem of the transition to turbulence in strongly confined systems is, at least at a conceptual level but also most often at a practical quantitative level, well understood in terms of temporal chaos and a small set of universal scenarios. This framework is, by construction, that of discrete systems introduced in Chapter 2. It can also legitimately be applied to continuous systems in case of strong confinement effects, by virtue of the distinction between driving and enslaved modes and the adiabatic elimination of the latter, thus offering a vast field where a detailed comparison of theory, tools and concepts, and laboratory or numerical experiments is meaningful. The next chapter is devoted to the study of the case when confinement effects are no longer sufficiently strong to restrict the effective dynamics to the interaction of such a small number of discrete modes.

4.6 4.6.1

Exercises

Homogeneous instability in a confined context

Consider a spatially extended system governed by &V

= T V -t a x x V - V 3 .

1) Linearized problem: determine the growth rate of Fourier modes a exp(ilcz) with infinitesimal amplitude in a laterally unbounded system and obtain the marginal stability condition of the trivial solution v E 0. 2) Adding boundary conditions v(z) = 0 at both ends of the interval [0,7r], check that one can take eigenmodes in the form X , = sin(nz) and determine their growth rate as a function of n.

3) Find the condition on T such that only the first mode is excited, while others remain strongly stable. When this is the case, assuming v(z,t)= A , ( t ) X I ( x ) ,determine the equation governing A1 within a firstharmonic approximation. Show that the system experiences a supercritical fork bifurcation.

4) The cubic nonlinearity generates modes X,, n > 1. Find the equation governing As, its order of magnitude at steady state when T T M . 4.6.5

Excitable s y s t e m

Consider a dynamical system in the form

x

= F y ( X )= X - X 3 - Y

,

(4.63)

where Y is, for the moment, a parameter. Determine the fixed points of (4.63), their number and nature (stable or unstable) as Y is varied. Determine the potential from which this equation derives (see Exercise 4.6.3, above). Suppose now that Y is a dynamical variable of its own, governed by Y = E(X - T ) ,

(4.64)

where 0 < E is a constant and T the control parameter. Determine the character of the fixed point of system (4.63, 4.64) as a function of T , assuming that 1 - 3r2 # 0 and E l/&. In the first case, explain the shape of the relaxation oscillations observed after damping of the transient. See Fig. 4.25. In the second case, the system is said to describe an excitable medium such that, for initial conditions X > T , the relaxation of the system towards its fixed point occurs after a large pulse. [In a reaction-diffusion system where the reaction part is accounted for by (4.63,4.64),the system develops fronts separating excited regions from relaxed one]. 4.6.6

Stability and attmctors of a two-dimensional s y s t e m

Consider the system

x=--aX+Y+XY,

(4.65)

Y = -bY - X 2 ,

(4.66)

4 . Nonlinear Dynamics: from Simple to Complez

c--

\

-

173

-4

r=l .O, ~ = 0 . 0 5

r=0.2, ~=0.05

.

Fig. 4.25

Excitable system. Top: Phase portraits. Bottom: Time series of variable X

where a and b are constants. By computing the eigenvalues and eigenvectors of the system when linearized around the origin X = Y = 0, show that it remains locally stable (stable with respect to infinitesimal perturbations) as long as a > 0 and b > 0. Sketch the phase portrait of the linearized system for a < b. Consider now the system with its nonlinear terms and determine the fixed points as functions of A = 1 - 4ab. This quantity will serve as a control parameter in the following. Show that a saddle-node bifurcation takes place when A changes its sign. Find the eigenvalues and eigenvectors of the vector field linearized around each non-trivial fixed point. Sketch the phase portrait of the system for A < 0, A = 0, and A > 0 (take e.g. b = 1 and values of a that lead to simple numerical applications, in particular a = 1/4, 3/16). Observe that the stable manifold of the unstable fixed point that extrapolates its stable eigendirection is the boundary

174

Instabilities, Chaos and lhTbUhCe

between the attraction basins of the two stable fixed points when they exist. Notice that the nonlinearities profoundly modify the aspect of the phase portrait expected from the linear analysis close to the origin. Numerical simulation of (4.65, 4.66) is re~ommended.,~ 4.6.7

Bruxellator

The system obtained in Exercise 1.5.1 reads X = A - (B

+1)X+X2Y,

Y = BX - X2Y.

(4.67) (4.68)

Find its fixed points as functions of parameters A and B and discuss their linear stability properties. Show that the reaction can bifurcate from a time-independent state towards an oscillatory regime beyond some critical value B, to be determined as a function of A. Compute the eigenvalues a t lowest order in E = B - Bc Adapt the first harmonic approximation of Chapter 2, 52.3.1.2, p. 49 t o derive the amplitude equation governing the system beyond threshold. 4.6.8

Locking

In a permanent regime, an oscillator can be described using a phase variable 0 defined modulo 2n and governed by 4 = w where w is the angular frequency of the oscillator. Consider two such oscillators weakly but nonlinearly coupled, governed by:

el = w1 - f ( e 2- el), e, = W, - f ( e , - e,) . Justify the fact that function f has t o be periodic with period 2n. In the following, take f ( c p ) = iKsin(cp) and write down the system for = 1 5(01 0 2 ) and cp = ;(dl - 6,) (define 6w = ;(wl - w , ) ) . Determine the condition on K ensuring the existence of fixed points t o the equation for ‘p. Discuss the nature of the asymptotic regime when this condition is (is not) fulfilled. Qualitatively describe the phase intermittency regime that takes place when the existence condition just misses being fulfilled.

+

e

27For a discussion of the physical relevance of this exercise, to be reconsidered in chapter 6, see: 0. Dauchot et P.M., “Local versus global concepts in hydrodynamic stability theory,” J. Phys. I1 France 7 (1997) 371-389.

4. Nonlinear Dynamics: from Simple 4.6.9

to Complez

175

Logistic map (variants)

Consider the map

where a > 0 is the control parameter. Draw the graph of fa, find its fixed points and study their stability properties. In the following, consider the dynamics of a trajectory starting in the neighborhood of X i + ) > 0 for a close to a0 where the fixed point bifurcates. Setting a = a0 e , rewrite the nonlinear map for X' = X - X i + ) in terms of E 0. [Answer: Fig. 4.29. For r] = 0.27 the attractor is a smooth loop (section of a smooth torus corresponding to a two-periodic regime for the time continuous system). For r] = 0.48, the attractor is a singular curve with fractal structure generated by stretchings and foldings in a regime of developed chaos. These corrugations appear around r] = 0.40 which is thus close to the border of chaos.] 4.6.14

Permanent us. transient chaos

Consider one-dimensional maps in the form

and more particularly the tent map defined by 0 5 XI, 5 1/2

1/2

5x

k

51

Xk+l

=a

xk+l

= a(1 - x

x k

7

k ) .

1) Suppose first a = 2 and draw the graph of 7 and check that interval I[,-, [0,1] is invariant in the sense that it contains its image by 3. Then

find the fixed points of the map and show that they are unstable. Draw the graph of the iterated map For,i.e. X t-+ F ( F ( X ) ) .Find its fixed points and obtain from them a periodic orbit of F with period 2. Show that this

4 . Nonlinear Dynamics:

from

Simple to Complez

179

r------

2.5

-1.5'

-2

Fig. 4.29

-1.5

-1

-0.5

0

0.5

1

The Curry-Yorke map 9 = 2 and several values of 11.

orbit is also unstable. The orbit of an initial condition chosen at random is usually (with probability 1) chaotic. Compute its Lyapunov exponent using (4.37). 2) Suppose now that 1 < a < 2 and draw the graph of 3. Find the image II1 of 10 and the image 12 of 11, so that the definition interval of the map can be restricted t o 11. Show that all trajectories are unstable by computing the Lyapunov exponent.

3) Consider the case 2 < a. The interval 10 is no longer invariant. Draw the corresponding graph of 3 and find the set Jl of initial conditions that escape at the first iteration. Construct the pre-image J, of this set, i.e. such that T ( J , )= J1,to find initial conditions that escape in two iterations, and notice that it is made of two disconnected parts J; and Continue and observe that the initial conditions that do not escape belong to a Cantor set. When a = 3, the construction rule is exactly that of the triadic Cantor set. From the formula, compute the Lyapunov exponent of a trajectory starting right on this set (and remains in it). Such an unstable invariant set is called a chaotic repellor.

Ji.


Chapter 5

Nonlinear Dynamics of Patterns

The theory of deterministic chaos represents an important conceptual advance and offers an appropriate framework to account for experiments in confined geometry. However, it becomes rapidly inapplicable when the effective dimension of the dynamics increases, which is the case for extended systems (large aspect ratio). As already indicated, eigenmodes are then quasi-degenerate with wavevectors typically such that

C being the lateral extension (see p. 117). Many modes may be unstable close to the threshold. A large part of the interaction between them can be understood in terms of linear interferences accounting for spatial modulations brought to a regular uniform reference pattern (at the limit of a laterally unbounded system), e.g. parallel straight rolls for convection. At the nonlinear stage, focusing on the modulations directly leads one to the envelope formalism that gives a satisfactory account of confinement effects and defects at lowest order, a first step toward understanding spatiotemporal chaos. For simplicity, we mostly limit this presentation to the case of cellular stationary instabilities. The case of dissipative waves, just briefly introduced in 85.4.4, is left to more advanced studies. We consider first, in $5.1, the determination of uniform quasi-one-dimensional structures, periodic in a single space direction. Then we generalize the theory to the case of quasitwo-dimensional pat terns, with square or triangular /hexagonal plan-forms, in 85.2 and $5.3. Their most universal instability modes in terms of long wavelength perturbations are next investigated in 55.4. We conclude the chapter by 85.5 where we present some other modeling approaches that may help one analyze space-time chaotic regimes in extended systems.

181


182

5.1

5.1.1

Quasi-one-dimensional Cellular Structures Steady states

Consider an instability mechanism which, like plain Rayleigh-Bknard convection, generates stationary structures locally periodic in a single space direction, say x. At the linear stage we have:

the solutions of which are searched for in the form:

V = V exp(ikx

+ st)

(5.2)

which leads to the dispersion relation:

L ( i k , s; R ) = 0 .

(5.3)

We are particularly interested in the neighborhood of the critical conditions R M R,, corresponding to a marginal mode k = k, for which R e ( s ) = u = 0. (Zm(s)E 0 along a stationary instability branch.) At threshold, we have L(ik,, 0; R,) = 0, so that the linear problem (5.1) has a non-trivial solution:

V c = V, exp(ik,x)

+ C.C.

(5.4)

where V, accounts for the structure of the critical normal mode and C.C. means ‘complex conjugate’. The nonlinear problem extending (5.1) reads:

where hf(V,V )represents the higher order terms that were neglected in the linearization procedure. The notation suggests formally quadratic nonlinearities, as in hydrodynamics. The solution is searched for as an expansion in powers of a small parameter E :

v = EV1 + E2V2 + E 3 V 3 . . . ,

(54

and, as in the calculation of the period of nonlinear oscillators (Chapter 2, especially §2.3.2.3),the control parameter is also expanded:

R = R,

+ E R+~e2R2 + .

,

.

(5.7)

5. Nonlinear Dynamics of Patterns

183

Isolating the distance to threshold in the expression of the linear part on the 1.h.s. of (5.5) we can write:

C =L c

- (R-

R,)M,

(5.8)

where M is the opposite of the formal derivative of C with respect t o R evaluated a t threshold. Inserting expansions (5.6, 5.7) into (5.5) and taking (5.8) into account, we get a series of linear problems:

LV1= 0 CVz = RiMV1+ n/(Vi, Vi) , LV3 = RzMV1+ R1MVz n/(VZ,Vi) n/(Vl, V,)

+

+

(5.9) (5.10)

,

(5.11)

of the general form:

CVk = Fk.

(5.12)

The first problem is homogeneous. Since the critical conditions are fulfilled, it has a non-trivial solution V1 c( V,. The higher order problems (k > 1) are all inhomogeneous and depend on the solutions computed at previous orders (k' < k). Unknown free quantities Rk introduced through (5.7) are fixed by the condition that the r.h.s. of (5.12) do not contain resonant terms, exactly like in the Poincark-Lindstedt calculation, p. 55ff. At a formal level, let the relevant scalar product' be denoted by (...I...) and the adjoint Ct to 12 be defined by:

(WICV) = ( V l d W ) * , the conditions (Fredholm alternative) fixing the unknown parameters in (5.7) read:

(VIFk) = 0 .

v

where is the solution generating the kernel of Ct (CtV = 0). From (5.9) one gets V1 c( V, and when applied to (5.10), the condition for k = 2 yields R1. Once this condition is fulfilled, a particular solution can be found, to which one may add an arbitrary solution of the homogeneous problem to obtain the most general solution. To fix this solution uniquely, one may ask it t o be orthogonal to V,, so that it appears as a true correction to the first order solution (in the sense of the scalar product):

(VCIVZ) = 0 . lSee Appendix A, sA.3.2 for a reminder.

184


Once V2 is determined, it is reported in (5.11), where the sole unknown on the r.h.s. is R2, and so on. When R1 gives:

# 0, one can truncate the expansion at lowest order, which R = R,

and, after elimination of

E

+E R ~ ,

v =EV,,

between the two equations

so that the bifurcation is in fact two-sided, the solution exists for both < R, and R > R,, and its amplitude varies linearly. This situation was encountered in Exercise 4.6.3, Eq. (4.59) describing what was called a trans-critical bifurcation where two solutions exchanged their stability. Though non-generic, this situation was shown to occur in the absence of ‘ A -A’ symmetry.

R

In fact, it often happens that R1 = 0 for symmetry reasons. This is the case of Rayleigh-Bhard convection with symmetric top/bottom boundary conditions within the Boussinesq approximation. The ‘ A -A’ symmetry then results from the translation invariance by Xc/2 in the direction perpendicular to the roll axis. When R1 = 0, the lowest non-trivial truncation of the expansion is one order higher:

R = R, i- t2R2,

v = €VC+ 2v2,

and thus, neglecting c2V2when compared to EV,for we obtain:

V

21

E

sufficiently small,

k J ( R - R,)/Rz V ,

The bifurcation is now one-sided. Bifurcated states are to be found either for R > R, when R2 > 0 (supercritical bifurcation) or for R < R, when R2 < 0 (subcritical). Explicit calculation shows that Rayleigh-Bknard convection between good-conducting plates is supercritical. It is of course possible to continue the expansion and determine a more accurate solution by going t o next order k+1 since everything is known at order lc (Rk-1) or can be determined ( V k - 1 ) and that the compatibility condition contains only Rk as unknown.


5.1.2

185

Amplitude equation

We now turn to a variant of the same calculation that brings back a problem similar to that studied in Chapter 4, $4.1, reintroducing time in a way similar to the method of multiple scales, Chapter 2, $2.3.2.4. We consider the emergence stationary dissipative structure and make it explicit that the bifurcation is supercritical by defining a new small parameter E through:

R= R,+E~.

(5.13)

We again search the solution as an expansion:

v = &V1+ &2V2+ ... ,

(5.14)

but we no longer assume that it is time-independent. In order to account for this new feature we introduce a slow time scale tl in addition t o the natural time scale that we now denote as to by setting:

(5.15) The order of t l in E results from the choice (5.13) and anticipates the fact that, close t o the threshold, the growth rate of perturbations varies as R - R,. Here, since the instability is stationary, the action of a,, is trivial.2 present in C. With Let us come back to (5.5) and expand also the respect to system (5.9,. . . ), in addition to the assumptions R1 = 0 and R2 = 1 inherent in (5.13) the first important modification enters Eq. (5.11) where a term at, appears. We can thus rewrite it as:

at

LV3 = M V i

+ M(V2,Vi) + N ( V i ,V2) - Q &,Vi ,

(5.16)

where Q is the opposite of the operator obtained by differentiating 12 with respect to at formally. The solution at order E reads

V1 = AIVc exp(ilc,z)

+ C.C.

is now a function of the slow variable t l . At order E ~ Eq. , (5.10) is left unchanged with the introduction of time. The compatibility condition that determined R1 in the previous approach, is now trivially fulfilled by assumption. Eq. (5.10) contains non-resonant where

A1

2Things would be different for an oscillatory instability, in which case we would have -iwc, but the approach can easily be extended owing to what we learned in Chapter 2.

ato =

186


terms of the form exp(inkcz) with n = 0 and n = f 2 issued from the evaluation of n/(Al exp(ik,z) + c.c., A1 exp(ik,z) C.C.). A particular solution can thus be found in the form:

+

where the first subscript indicates the order in E and the second one the harmonic generated by the nonlinear couplings. A solution of the homogeneous problem, A2Vc exp(ik,z) c.c., must be added to this particular solution in order to obtain the full solution at order e2. At order c 3 , it is easily seen that Eq. (5.16) contains a certain number of resonant terms coming from the evaluation of n/(V2, V1) +n/(V,, V2). Because V2,partcontains harmonics 0 and f 2 and V1 harmonics f.1, these terms contain harmonics 0 f 1, i.e. f l , and f 2 f 1, producing f 3 and f l . Other resonant terms come from R2V1 (with R2 = 1 by definition), and Q at1V,. Instead of giving R2 as in the previous approach, the compatibility condition now reads:

+

One gets easily convinced that the second term is of the form IA1I2A1 by counting the powers of exp(ik,x), so that (5.17) effectively reads:

where TO,a , and g are constants that can be evaluated by computing the scalar products in (5.17). Returning to notations introduced in Chapter 3, especially through (3.23), p. 84, coefficient a can be identified with R;' since the linear growth rate was defined there as u = T;~(R - Rc)/Rc. For a stationary instability, it is easily shown that if the system is symmetrical under the change x I+ -2, then (5.18) has real coefficients since this symmetry implies symmetry under complex conjugation: Aexp(ik,x) I+ A*exp(-ik,z). This is no longer the case for oscillatory instabilities and dissipative waves which are much more complicated in this respect, see 55.4.4 below. The expansion can be continued. It is then observed that the freedom introduced by the Rk in (5.6) for the computation of the time-independent solutions (5.7) is now replaced by the introduction at each order of the successive amplitudes Ak. The Redholm alternative now governs these Ak as functions of the slow time tl.


187

At lowest order we have just A = &A1.Coming back to the natural time

t using (5.15), observing that all terms of (5.18) are of order E ~ we , get: n&A = a ( R - R , ) A - glAI2A

(5.19)

called the amplitude equation. As announced, this result is in line with the argument developed at the beginning of Chapter 4, and especially with that leading to (4.7).

5.2

Dissipative Crystals

We now generalize the previous calculation €or the case of systems that are isotropic in the (z,y) plane perpendicular to direction t singled out by the instability mechanism. The linearized problem now reads:

C(V1, at; R ) V = 0 ,

(az,8,)

V1

(5.20)

and, owing to the orientation degeneracy, its solution can be searched for as a superposition of plane waves with wavevectors in different directions v 1

=c

+ C.C. ,

v k j eXp(ikj*Xh)

(5.21)

kj

with lkjl = k, for all j . The expansion in powers of E is performed as in $5.1.1 and again leads to (5.9, . . .). Let us work sequentially as before. Though the equation at order c2 remains formally the same as in the one-dimensional case, the twodimensional character of the probIem implies novelty in the determination of resonant terms: nonlinearities h/(V1,V,) that were then only able to generate the non-resonant harmonics 0 and 2, can now produce resonant combinations as shown in Figure 5.l(b). We can now rewrite (5.10) in the form

+

so that when kjl kj,, falls right on the critical circle, we must compensate this term with a term in R1 # 0 for some well chosen kj. In the case of formally quadratic nonlinearities, it is thus generically expected that solutions at order c2 exist in the form of regular superpositions of three wavevectors at angles 2 ~ / 3(and their opposites). Such solutions bifurcate

188


+

Fig. 5.1 Spatial resonance at second order: (a) Non-resonant combination k = kl kz with Ikl # k,. (b) Resonant combination k = kl kz with Ikl = k,. (c) Regular superposition of three wavevectors making a three-branch star at 2 ~ / forming 3 a resonant set with their opposites (dashed arrows).

+

trans-critically as already shown and only special circumstances can suppress them by canceling the scalar products involving the last terms on the r.h.s. of (5.22) for symmetry reasons (e.g. the top-bottom symmetry in convection). Let us suppose that the nonlinearities do not generate resonant terms at second order. The solution is still a priori made of a superposition of linear modes (5.21). So, let us consider two pairs of wavevectors f k l and f k z and start with:

The special solution at second order then formally reads:

Inserting these expressions in (5.11), we obtain terms with space dependences in the form:

so that we cannot avoid the generation of resonant terms through the relation:

kji

+ kjri + kj,,, = k ,

with k lying right on the critical circle, as shown in Figure 5.2. The difficulty in the computation only comes from the fact that this relation can be fulfilled in many ways. Among all possible combinations, those involving a single pair of wavevectors are immediately identified,


189

-+

--- - -

(1) = 1+1-1 (2) = 1-1+1

(a)

(b)

(4) = 1-2+2

(6) = -2+2+1

(5) = 1+2-2

(7) = 2-2+1

e (3) = -1+1+1

(8) = -2+1+2

(9) = 2+1-2

Fig. 5.2 Spatial resonance at third order: (a) An example of resonant superposition; (b) N pairs of wavevectors making an angle ?r/N between nearest neighbors, here with N = 3. (c) Combinations with a single pair of wavevectors f k (rolls). (d, e) Combinations with two pairs of wavevector forming a parallelogram, degenerate (d) or not (e).

which we already had in the quasi-one-dimensional case. But other combinations with nontrivial contribution to Rz are readily discovered, Fig. 5.2(a), whose contributions further depend on the angles made by the wavevectors pairs in a quantitative way. In general one is first interested in the coefficient RL" associated to superpositions of N pairs of wavevectors forming angles of 7rlN between neighbors, Fig. 5.2(b). Apart from stability considerations to be examined in 35.3 below, superpositions with an arbitrary number of pairs can be considered. However, obtaining simple periodic patterns implies either N = 1 for rolls, N = 2 for squares, or N = 3 for hexagons as demonstrated for standard two-dimensional crystal^.^ Superpositions with more than three regularly disposed pairs of wavevectors in general form multi-periodic patterns called quasi-crystals. The latter may degenerate into periodic super-lattices with large periods when certain commensurability relations are fulfilled. See Chapters 5 and 6 of [Rabinovich et al. (2000)]. The parallel is complete with periodic locking 3See, Chapter 13 of L.D. Landau & E.M. Lifshitz, Statistical Physics (ButterworthHeinemann, 1980).

190

Instabilities, Chaos and Tbrbulence

of quasi-periodic regimes studied in Chapter 4, 54.2.3. A natural extension of the temporal setting would suggest the existence of chaotic crystal^.^

5.3

Short Term Selection of Patterns

What has just been said relates only to the existence of stationary nonlinear dissipative structures but the approach leading t o the amplitude equation (5.19) for a roll pattern, characterized by a single complex amplitude A , can be reproduced for more complex patterns with superpositions of plane waves exp(ikj.xh), with amplitudes A, and their complex conjugates. Symmetry and resonance considerations (see Exercise 5.6.1) lead t o phenomenological amplitude equations generalizing (5.19). Assuming formally cubic nonlinearities (or formally quadratic but such that relevant resonances at second order are killed for symmetry reasons), one obtains:

where coefficient go and gjjl respectively accounts for interactions of type (c) and (d, e) displayed in Figure 5.2. When formally quadratic interactions do not kill resonances at second order, sets of three pairs of wavevectors, k j , j = 1 , 2 , 3 , such as in Figure 5.l(b) have to be considered with the corresponding amplitudes Aj, j = 1 , 2 , 3 . The resonance condition k l = -kz - k3 then implies the presence of a term AZA; in the equation for A l , which leads to:

and two other equations obtained by circular permutation of the subscripts. Coefficients g, p and j in (5.24) are to be determined from a detailed nonlinear calculation or just introduced through a phenomenological argument. Regular configurations correspond to specific fixed points of the amplitude equations (5.23) or (5.24) with IAj( = A for all j . Patterns selected by nonlinearities can be discussed from the stability of these fixed points in the strictly temporal setting of Chapters 2 and 4. The case of squares is examined in Exercise 5.6.1. 4A.C. Newel1 and Y. Pomeau, “Turbulent crystals in macroscopic systems,” J . Phys. A: Math. Gen. 26 (1993)L429-L434.


5.4

191

Modulations and Envelope Equations

The previous analyses all referred t o uniform cellular structures with wavelengths equal to the critical wavelength A,. By assumption, confinement effects are weak in extended geometry and thus may prove unable t o control the development of such “ideal” structures. “Natural” patterns that develop are therefore usually disordered, with local vectors k such that Ikl x k, but with slowly variable lengths and/or orientations not everywhere perfectly aligned with the directions defining a unique underlying reference pattern. Defects can also perturb the regular ordering of the individual cells. An example from convection in a large Prandtl number fluid was displayed in Figure 3.10, p. 98. Imperfect patterns are usually called textures. The problem is approached within the framework of so-called envelope equations, which adds a spatial meaning to the temporal dimension of the amplitude equations introduced above, thus making possible the description of slow modulations to regular patterns and universal instabilities attached t o them. The derivation again starts with a solution to the nonlinear problem as a superposition of modes but with amplitudes that can be slowly varying in time and space, now called envelopes. Through the derivation process, most specificities of the primary instability are rubbed out, so that the result is expected t o bear a universal content: all patterns with the same symmetries behave in the same way. Here we restrict ourselves to a heuristic approach5 mostly using symmetry arguments and we rest the discussion on properties of the linear dispersion relation in the neighborhood of the threshold as discussed earlier in Chapter 3, 53.1.6, p. 84. We consider first the case of a one-dimensional cellular instability before proceeding t o several extensions. 5.4.1

Quasi-one-dimensional cellular patterns

At distance r = (R-Rc)/R, from threshold R,, the growth rate of a normal mode with wavevector k = kc Sk is generically given by (3.26), i.e.:

+

700

= T - t o S2 k 2 .

(5.25)

5For technical details consult the seminal paper by A.C. Newell, “Envelope equations,” Lectures in Appl. Math. 15 (1974) 157-163; or else: S. Fauve, “Pattern forming instabilities” in [GodrBche and Manneville (1998)l; A.C. Newell, Th. Passot, and J . Lega, “Order parameter equations for patterns,” Annu. Rev. Fluid Mech. 25 (1993) 399-453; [Cross and Hohenberg (1993)] or else [Rabinovich et al. (2000)l.


192

Slightly above threshold (0 < r 0) in the band k E [k, - A k , k, - A k ] with A k = [i’fi.The space modulations are thus slow when compared t o variations a t the scale of the wavelength A, ( A k / k , > / ~ 8, ) N 0(r1/’) for T 1, the calculation developed in Exercise 5.6.1 shows that rolls are preferred locally." Exercise 5.6.3 then shows that a system of rolls with a wavevector too far from k, also becomes unstable against the formation of rectangles owing to the growth of a system of rolls with wavevector kc but at right angles with it. This is the cross-roll instability, and the way it is shown to exist implies its universal character: all roll patterns in rotationally invariant media may experience it. On the other hand, owing to their local stability properties, roll systems at right angles may coexist in different but contiguous regions of space if their wavevectors are sufficiently close to kc, forming a stationary grain boundary, see Figure 5.4 (left). A grain boundary parallel to the y axis is obtained by solving (5.32, 5.33) in 08

,

,

,

,

,

,

,

,

,

Fig. 5.4 Left: Roll system with a grain boundary. Right: Solution for the corresponding amplitudes.

loIn essence, when p < 1, everything occurs as if p = 0,the two systems of rolls ignore each other, and grow, so that squares are obtained everywhere in space; when p > 1, they cannot but feel each other and one kills the other, at least locally.

196


the special case a, 0 which leaves one with a differential system in x. For example if both underlying wavevectors are equal to k,, one gets:

where the primes indicate differentiation with respect to x. The solution illustrated in Figure 5.4 (right) has here been rapidly and accurately obtained by searching it as the asymptotic solution of a partial differential problem in x and t using a simple numerical scheme of the kind described in Appendix B, 5B.2.1. The mathematical solution can however be obtained by analytical means (matched asymptotic expansions).” The envelope formalism just introduced can account for the essentials of the dynamics of scalar textures such as those observed in convection a t large Prandtl numbers. In particular, it explains the orientation of rolls at lateral boundaries, the presence of curvature and well ordered grains and other defects such as those present in Figure 3.10, p. 98. All this follows from the fact that the NWS equation (5.29) derives from a potential in the sense of Chapter 2, $2.1.3, p. 33, i.e. can be written as (5.34) where the right hand side is a notation representing the functional differentiation of

with respect to A*, with g E R, hence E R. Functional differentiation in (5.34) is understood in the sense of variation calculus. In the change A t-+ A + 6A: 0 variations of IAI2 and IAI4 expand as AdA* A*6A and 21A12(A6A* A’dA), which immediately gives the corresponding terms in (5.29) upon isolation of terms in 6A*; terms involving derivatives need a slightly more complicated treatment; for example, the variation of aXAaxA*gives axAazSA* ax6AazA* but ax6A* is not independent of 6A* so that an integration by part has t o be performed to isolate the latter, which yields -6A*axx A plus boundary

+

+

+

“P. M. and Y. Pomeau, “A grain boundary in cellular structures near the onset of convection,” Phil. Mag. A 48 (1983) 607-621.


197

terms; the other terms can be treated in the same way (two integrations by part for the terms arising from l13yyAlz)so that all the partial derivatives of (5.29) with respect to space are also recovered; 0 boundary conditions (5.28, 5.30) for rolls parallel or perpendicular to the lateral boundaries may be used to cancel boundary terms arising from integrations by parts in case of a rectangular domain; otherwise contributions from the boundary term can simply be neglected when compared to bulk contributions in the general case. The system then evolves so as to minimize the potential

a t )=

G(&,

Y, t ) ,A * ( Z , Y,t))dx dY

over the domain considered since $G 5 0. Using the same argument as in 52.1.3, one obtains that the solution that achieves this local minimum is time independent. On this basis, selected patterns are those corresponding to stationary solutions of (5.29) at local minima of 6 (local maxima or saddles are unstable solutions). Favored textures can be found by comparing the contributions of the different causes of inhomogeneity (lateral walls, defects) to the potential increase with respect to the uniform solution. In this context, the slow residual time dependence often observed in experiments can be attributed to higher order terms omitted in (5.29) which only holds at lowest significant order, at least in the scalar case. However, the interpretation of space-time chaos (weak turbulence) is at any rate more complex as soon as one leaves this simple framework, in particular for convection at low Prandtl numbers for which the most relevant field is no longer the temperature but the velocity. 5.4.4

Oscillatory patterns and dissipative waves

Up to now, we have considered only stationary instabilities. The approach in terms of envelope equations extends also in the more difficult case of oscillatory instabilities, w, # 0, in particular for waves when Ic, # 0. Difficulties arise from the fast time dependence, drastically destroying the relaxational property by introducing complex coefficients in the evolution equations, as in the strictly temporal setting of standard Hopf bifurcation, Chapter 4, 54.2.1.2, p. 128, and not mildly by higher order corrections as above. Here, we just give a few results practically without derivation, inviting the reader to consult references mentioned in Note 5. As in the stationary case, the primary role is taken by the linear dispersion relation introduced in

198

Instabilities, Chaos and Ilrbdence

Chapter 3, $3.1.6. Considerations about resonances developed in Chapter 4, $4.1.3,are then used to deal with nonlinearities directly written as normal forms relevant to the case at hand. For an oscillatory instability with kc = 0 or for a dissipative wave with k, # 0 but in a reference frame moving at the group velocity (see p. 85), we have:

This equation accounts for the space unfolding of the Hopf bifurcation (4.16) as derived from the dispersion relation of the corresponding instability (3.26, 3.27), p. 84. Coefficient 6 is the critical pulsation in units of TO,i e . w, = 5/70; it can be eliminated by changing to a ‘rotating frame’ A ( z ,t ) H A ( z ,t )exp(-iwot). The next coefficient, a,is a measure of linear dispersion in units of the natural coherence length ti. Finally, ,B is the nonlinear dispersion coefficient rescaled by g, which accounts for saturation effects by nonlinearities. Upon further rescaling of time, length, and amplitude, one obtains the universal form of the so-called complez Ginzburg-Landau equation (CGL):12

&A = A + (1 + ia)d,,A

- (1

+ i,B)IAI2A.

(5.36)

This formulation is not restricted to the one-dimensional case introduced here but can be extended to higher dimensions, a,, simply being replaced by the Laplacian when the physical system is isotropic in space. The structure of the equations governing one-dimensional waves propagating in opposite directions (complex envelopes A1,2) can be obtained in the same way by symmetry and resonance considerations which lead to:

through appropriate rescalings. Notice however that, since the first order partial derivative 8, is expected to be of order r1I2 before rescaling, the unscaled group velocity must also be of order r1I2 ( i e . small enough), t o insure the consistency of the set of equations at order r3I2. l 2W. van Saarloos, “The complex Ginzburg-Landau equation for beginners,” in [Cladis and Palffy-Muhoray (1995)l.


199

Solutions with A2 = 0 and A1 # 0 (or the reverse) account for propagating waves moving to the right (or left), i.e. z A l ( a : - v t ) (or Aa(a:+vt)). By contrast condition /All = lAnl describes a superposition of right and left waves forming a standing wave. A calculation analogous to that in Exercise 5.6.1 shows that the system prefers propagating waves when Jpll> 1 (one wave L‘feels’’the presence of the other and “kills” it) and standing waves when Ip‘I < 1 (each wave does not L‘fee”’the presence of the other much and therefore ‘Laccept”cohabitation). When propagating waves are preferred, defects similar to grain boundaries may separate homogeneous domains with opposite kinds of wave in each. These defects are called sources or sinks according to whether the waves travel away from the defect or toward it.

5.4.5

Universal long-wavelength instabilities

One of the interests of the envelope formalism is to offer a framework for the study of universal secondary instabilities of dissipative structures linked to the symmetry properties of patterns at the limit of a laterally unbounded system. For stationary cellular structures, two such secondary modes are the Eckhaus instability against local compression/expansion and the zigzag instability against local torsion of the rolls, as sketched in Figure 5.5. These two universal modes are called phase instabilities because they relate to modulations of the position of the cells. Within the envelope approach, the solution is searched for in the form A = IAlexp(i+), and while the modulus JAldescribes the intensity of the response to the primary instability mechanism, the phase serves to specify the absolute position of the pattern in the laboratory frame. The universal instability linked to the

Fig. 5.5 Initial aspect of perturbation associated to the Eckhaus instability (left) and the zigzag instability (right).

200


amplitude is the cross roll instability mentioned earlier, p. 195, and further studied in Exercise 5.6.3. Contrasting with the amplitude that has a finite relaxation time, the phase is dynamically neutral as long as the solution is uniform. A slightly irregular pattern can be described by a phase modulation which decays when the pattern is stable or gets amplified when it is unstable. Typically, at least for scalar dissipative structures, the phase of the envelope is governed by a diffusion equation:

at4 = D,,azz4+ DLayy4 ,

(5.37)

where Dll and D l are diffusion coefficients along the wavevector of the structure, or perpendicular to it. As long as these coefficients are both positive, the phase perturbation relaxes and the roll pattern is stable against (infinitesimal) phase perturbations. The instability is observed when one of these coefficients changes its sign. The Eckhaus instability corresponds to Dll becoming negative, and the zigzag when it is D L . The phase diffusion coefficients depend on the wavelength of the underlying pattern. Exercise 5.6.2 is a first approach to the Eckhaus instability resting on (5.27), the zigzag instability would be studied in the same way using (5.29). The result is generally presented as a stability diagram in the parameter plane (6k,r ) where 6k measures how the wavevector of the underlying pattern departs from the critical wavevector and r is the control parameter. Figure 5.6 displays the results at lowest significant order. The Eckhaus instability is a side-band instability that develops “far from” k, and close to the marginal stability curve, in a region where the instability mechanism is not very efficient and the amplitude of the solution, A0 = d(r- t:bk2)/g, is small. “Preferring” a solution with a more optimal wavevector, the system amplifies the phase modulation by compressing certain regions and expanding others (Fig. 5.5, left). Supposing for example that the initial wavelength is too short, expanded regions will have a more favorable local wavevector while in compressed regions the local wavelength will be even shorter. The primary mode completely disappears at the location of the most compressed places since the amplitude of the primary mode drops to zero at the marginal stability curve. The pattern then loses a pair of rolls at such places and the wavelength further relaxes toward an increased more optimal value. If the obtained wavevector is still in the unstable domain, the process repeats. If not, it stops and the system evolves through diffusive relaxation toward a uniform pattern with a wavelength inside the stable domain.


201

Fig. 5.6 Stability diagram of roll patterns against universal phase modes of compression/expansion (Eckhaus) and torsion (zigzag). As obtained from equation (5.29) valid at lowest order, the marginal stability curve is given by T = < i S k 2 . Rolls are unstable against the Eckhaus mode in the region (idk2 < T < 3503dk2. The zigzag instability domain is for 6k < 0. Stable rolls have wavevectors in the domain 0 < dk, T > 3503dk2.

The zigzag instability sets in when 6k < 0, i.e. for wavelengths larger than critical. This can also be easily interpreted by noticing (Fig. 5.5, right) that, when measured perpendicularly to the local axis of the rolls, the wavelength is shorter than that measured along the 2 axis by a factor equal to the cosine of the angle between the local periodicity direction and the x axis. The instability again develops so as to make the (too long) initial wavelength shorter and thus closer to the critical value. By contrast with the case of the Eckhaus instability which changes the number of rolls, the zigzag instability saturates as it amplifies. The process ends in wide regularly spaced domains of ‘zig’ and ‘zag’ straight rolls connected by barrow bands of strong bending. Up to now, we have only considered the initial development of these universal phase instabilities and just sketched the ultimate fate of the unstable state. Analytically, one can go a little further and complete the phase diffusion equation (5.37) by appropriate nonlinearities. In this context, let us mention the Kwamoto-Sivashinsky equation13 (KS) to be used in the numerical experiments described at the end of Appendix B. This equation describes the dynamics of the phase of nearly uniform solutions to the complex Ginzburg-Landau equation (5.36) in a narrow neighborhood of the threshold of the Benjamin-Feir instability. 13Y. Kuramoto, “Phase dynamics of weakly unstable periodic structures,” Prog. Theor. Phys. 71 (1984) 1182-1196.


202

On general grounds this instability, which closely corresponds t o the Eckhaus instability of one-dimensional stationary patterns, develops when the phase diffusion coefficient D = 1 ag becomes negative (Newell’s c r i t e r i o n , see Note 5. By symmetry considerations one can check that the equation governing the phase, at most quadratic in and up to order four in ax,reads:

+

+

at4 = DaXX4 - K a x x X x 4 + 9 0 ( & 4 )+~ g i ( a X 4 ) ( & x z 4 )

+ 9 2 ( a x ~ + )+~ 9 3 ( a ~ + ) ~ ( a x x + )

(5.38)

where each coefficient can be derived from those in (5.36). In addition t o D = 1 a,8, one gets14 K = ;a2(1+ p 2 ) > 0, 90 = P - a, 9 1 = 2 9 2 = a93 = -241 +p). In practice, for D M 0 but negative, the order of magnitude of the space gradient is fixed by the competition between the two linear terms and all nonlinear terms except the first can be neglected. After appropriate rescalings, Equation (5.38) can be reduced to the KS equation which reads in universal form:15

+

or, after differentiating it with respect to x and setting $ = ax+: at$

+$ax$ +

8XX$

+ ax,,,$

= 0.

(5.40)

The nonlinearity +a,$ present in this last expression is reminiscent of the advection term of hydrodynamics. It already appeared in the B u r g e r s equation (Chapter 1, Exercise 1.5.3) producing shocks (Exercise 5.6.4, part 1) and the Korteweg-de Vries equation producing solitons (Chapter 2, Exercise 2.5.6). The KS equation in one or the other form, (5.39) or (5.40) is a particularly simple model of phase turbulence.16 As long as D remains small, the modulus of the envelope is enslaved t o the phase gradient, stays close to its nominal value Ao, and therefore remains bounded away from zero. This is no longer the case when the phase instability is more intense. The field IA( then revolts and recovers a 1 4 J . Lega, Defauts topologiques associ6s B la brisure d’invariance de translation dans le temps, PhD Dissertation, Nice University, 1989 (in French). 15The equation also appears in problems of front propagation as shown by G.I. Sivashinsky, “On self-turbulisation of a laminar flame,” Acta Astronautica 8 (1979) 569-591; hence the joint names for the equation. laFor a brief review with references, see: H. Chat6 and P. M., “Phase turbulence,” in [Tabeling and Cardoso (1994)l.


203

dynamics of its own. It then explores a larger range of values that extends down to zero. At places where IAl = 0, the phase is no longer defined and phase defects nucleate, with 2~-jumpsof 4. The CGL equation then enters new regimes whose precise nature depends on the value of cly and ,b in (5.36) and the space dimension. Strong space-time chaos called defect turbulence then sets in. In two dimensions, it is characterized by the permanent birth of defects in pairs, that further dissociate and move around before merging.17 The theory of the transition to chaos is rooted in the idea of dimensional reduction leading to effective dynamical systems in terms of ordinary differential equations. To summarize this section, one can say that the envelope formalism is the required adaptation of this idea when confinement effects are too weak to legitimate the approach in terms of isolated modes and corresponding discrete amplitudes. Pattern selection strictly relies on this reduction for uniform solutions. Defects and universal secondary instabilities involve modulations to some ideal reference situation. As long as the system remains sufficiently close to a stable regularly ordered pattern, the relevant instability modes relates to the phase of envelope and a supplementary reduction is possible by adiabatic elimination of the envelope modulus. Localized defects also play a role, either because they are present in the initial conditions (when the pattern emerges) or as the result of secondary instabilities that do not saturate. Sometimes tools borrowed from the theory of dynamical systems can still be used, e.g. to determine special solutions to the envelope equations. It should also be noticed that, while the envelope formalism can be made rigorous within the framework of multiple scale methods, an interesting alternative is the derivation of generic models by phenomenological arguments resting on resonance and symmetry considerations. Further, the numerical simulation of such models has proved crucial to the understanding of space-time chaos. Such simulations usually do not require considerable investment and the reader is encouraged to practice them using the simple numerical methods presented in Appendix B, in order to get a personal intuition of the problem while taking advantage of the published material listed in, e.g. [Rabinovich e t al. (2000)] or to be found on the Internet. 17P. Coullet, L. Gil, J . Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62 (1989) 1619-1622. A complete phase diagram is given in the one-dimensional case by H . Chat6, “Disordered regimes of the one-dimensional complex Ginzburg-Landau equation,” in [Cladis and Palffy-Muhoray (1995)l.

204

5.5


What Lies Beyond?

Before closing this chapter, let us point out the interest of fully analogical approaches to the modeling of extended systems, useful when the minimum space-time coherence necessary to apply the envelope formalism is absent from the system. As a matter of fact, when the space-time coherence is limited, there is some advantage to consider the continuous system as an aggregate of subsystems coupled to each other. The local dynamics is accounted for at the scale of the subsystem while the space extension arises through the coupling between the subsystems usually arranged at the node of a lattice. A system can thus be discretized at several levels. At first, one can just discretize the physical space and get lattices of differential systems, e.g. lattices of Hopf oscillators or Lorenz systems. These systems are then coupled by some rule. For example, nearest-neighbor diffusive coupling of identical Hopf oscillators in one dimension would yield

where subscript n indicates the space position of an individual oscillator and the discrete version of the diffusion operator, here Da,, is easily recognized. Next step, time can also be discretized, which gives coupled map lattices. Time and space being indicated by superscript k and subscript n, respectively, in one dimension and again for diffusive coupling, one will start with

which is preferable to the seemingly more straightforward formulation:

x;+’ = F (xi)+ D [x;,’ - 2x; + x;-,] , since it is immediately checked that the latter may not preserve the invariant domain of each subsystem while the former does (if X E V implies F ( X ) E ID, then X: E 2) implies Xk+l E V for all n). The appropriateness of the model relies entirely on the skill of the modeler while choosing the local map and the type of coupling.’* Up to now, the local phase space was still a continuous set. One can make a last step by considering cellular automata where each subsystem l8See for example [Kaneko (1993)] and for a specific application H. Chat6 and P. M., “Spatiotemporal intermittency” in [Tabeling and Cardoso (1994)].


205

can be in one of a finite set of states. Boolean automata have states labelled by bits 0 and 1, with evolution rules functions of the configuration of the neighborhood of each cell. Such systems can already have complicated dynamics in spite of very simple definitions [Wolfram (1986)]. Lattice gas automata are automata specially adapted to hydrodynamics. They describe fluids at the level of individual molecules but governed by simplified evolution rules. “Living” at the nodes of a regular lattice, the molecules can be in one of a discrete set of motion state (speed and orientation), jump from node to node and change their state of motion when they meet at some given node according to rules given in collision tables. At first, when massively parallel computers began to appear, they were presented as an alternative to the direct numerical simulation of hydrodynamics equations. Nowadays, they are rather considered as useful models when the local dynamics is rich or complicated (chemical reactions), or when the boundaries have a complex topology or special physical properties (surface catalysis). An interesting review of lattice gas automata as applied to complex hydrodynamics is [Rothman and Zaleski (1997)l. The present chapter was mostly devoted to stationary patterns in weakly confined systems, with only a few words about waves. The latter will be central to the study of open flow instabilities in the next chapter.

5.6

5.6.1

Exercises

Pattern selection

Consider patterns described by superpositions of plane waves:

i [ A j exp(ikj .xh)+ c.c.],

V= j

namely, rolls with a single pair of wavevectors and squares with two pairs at right angles. The wavelength of the participating modes is Ac = 27r/kc. 1) Show that translational invariance along wavevector kj = IC& (unit vector xj), xj I-) xj x j ( O ) , j = 1 , 2 , implies a (gauge) invariance for the corresponding complex amplitude Aj I-) Aj exp(@j(’)). Taking these symmetries into account, show that the supercritical bifurcation towards a uniform pattern is governed at lowest order by

+

dd t A 1 - rAl -g11(A1(2A1 - g 1 2 ( A 2 ( 2 A 1

(5.41)

%A2 = rA2 - g221A212A2 - gzi1AiI2Az

(5.42)

206


with gij E R. Use the rotational invariance to reduce the number of interaction coefficients to g (self-interaction) and 9’ = pg (cross-interaction). Show that g can be eliminated by appropriate rescaling of A1 and Aa. 2) Determine the potential G(A1,A ; , A 2 ,A ; ) from which (5.41, 5.42) derives in the sense of (5.34), here simply

and conclude that permanent regimes are time-independent.

3) Find all fixed points ( A l * ,AZ,) of system (5.41, 5.42) and, apart from the trivial solution, give their explicit expression and physical significance (take advantage of the gauge invariance to specify the phase of the envelopes so as to have real amplitudes). 4) Compute the value of the potential and study its curvature at the different fixed points to guess their stability properties. Then perform the explicit stability analysis by inserting Aj = Aj, aj in the system and linearizing. Even though the Aj, are real quantities, the most general perturbations are not, thus take aj = uj ivj. Show that the fourth order linear system obtained splits into two subsystems, producing two neutral modes (explain their origin) and two non-trivial modes that can be stable or unstable, according to the value of p and the considered fixed point. Use the result to interpret phase portraits displayed in Figure 5.7. What happens when p = l?

+

+

Fig. 5.7

Phase portrait of system (5.41, 5.42) for p = 0.80 (left) and p = 1.25 (right).

207


5.6.2

Eckhaus instability

Consider a roll pattern described by an envelope evolving according to:

&A = r A + d x x A- IAI2A. The Eckhaus instability relates t o the phase

(5.43)

4 of the complex field A.

1) Determine amplitude of a phase-winding solution corresponding t o uniform rolls with wavevector k = k, Sk, A ( x ,t ) = A(') exp(i 6k x). The stability of this solution against long wavelength perturbations (phase modes) is studied by assuming that the prefactor of exp(i 6k x) can be a function of space and time. From (5.43), derive the equation governing A ( x ,t ) defined through A ( z ,t ) = d(z,t )exp(i 6k x). 2 ) Insert A(z, t ) = A(') + a ( z ,t ) in this new equation, and show that the linearized equations governing the real and imaginary parts of a(x,t) = u(x,t ) iv(z,t ) read:

+

+

atu = -2(r - bk2)u 8tv

+ axxu- 2 b k d x v ,

+ aXxv.

= 26kdXu

Derive the dispersion relation for Fourier normal modes with growth rate s and wavevector q taken in the form u = .iiexp(s t ) cos(qx) and v = fiexp(st)sin(qx). 3) Show that the roots are real and that the phase-winding solution is unstable when

2(r - 36k2)

+ q2 5 0 ,

derive from this that the instability occurs first for q and recover the result displayed in Figure 5.6. 5.6.3

+ 0 (long wavelength)

Cross roll instability

In the conditions of Exercise 5.6.1, assume p > 1 (rolls preferred t o squares), consider a phase winding solution with k , = k, + 6k, and study its stability against rolls at right angles with k, = k,. From (5.32, 5.33), show that: $1

= TAl

+ &,A1

- g(IA1I2

= TAZ- g(lA2I2 is an appropriate starting point.

+ pIA2I2)A1,

+ pIA1l2)A2 ,

208


Show that the condition for instability has the same form as the Eckhaus condition but with a different prefactor to be determined. Compute the value of p that makes the cross-roll instability more dangerous than the Eckhaus instability and interpret the limit p + 1. 5.6.4

5.6.4.1

Dynamical systems and nonlinear waves

The Burgers equation

Consider the Burgers equation:

atv + vaxv = v a x x v . 1) Show that it is invariant through a Galilean change of frame (i.e. that the equation for V(E,t) with 5 = x - V t and V = v - V is the same as that governing v ( x ,t ) ) . (The corresponding symmetry of the NS equation is discussed in Note 9, p. 239.) 2) Derive the differential equation governing a solution that propagates at speed V without deformation (5 = z - V t is the sole independent variable) and integrate it once with respect to x. Consider a solution such that vh for 5 + f m , derive its speed V from the first integral just obtained, and derive the analytical shape of the corresponding solution. [Solution: a hyperbolic tangent.]

5.6.4.2 Kuramoto-Sivashinsky equation Consider variant (5.40) of the KS equation:

atv + vaxv

+ axxv + axxxzv= 0 .

1)Derive the linear dispersion relation governing infinitesimal perturbations 0. Show the instability of those belonging t o a range of q to be determined.

6v = &J exp(st + iqx) to the trivial solution v

2) Write down the ordinary differential equation in 5 = x - V t governing a solitary wave solution propagating without deformation in a frame moving at speed V and derive a first integral. By identification, find a solution in the form

V = atanh(n5) +,Btanh3(n5).

and compare the value of n with the wavevector qmax of the perturbation with maximum growth rate Smax.

209


5.6.4.3 Flow down an inclined plane The Benney equation is a partial differential equation governing the long wavelength perturbations to a fluid film of uniform thickness flowing down an inclined plane. After appropriate rescalings of length, time, and film thickness, one obtainslg

+

8th + $8, [h3 (ah6- h3)d,h

+ h3d,,,h]

=0

where a is the control parameter. 1) Derive the linearized equation governing infinitesimal perturbations t o a time-independent uniform solution h = ho. Show that solution h ho = 1 is linearly unstable for a > ac = 1.

=

2) A hydraulic jump is a solution such that h + 1for x -+ --oo and h + h, for x + +m. A solitary wave precisely corresponds to h, = 1. Write down the ordinary differential equation in 5 = x - V t governing a solution that propagates without deformation in a frame moving at speed V, integrate this equation to find a first integral and determine the value K of this first integral as a function of c when x -+ -m. Then derive the relation between V and h, < 1.

l9See: A. Pumir, P. M . , Y. Pomeau, “On solitary waves running down an inclined plane,” J . Fluid Mech. 135 (1983) 27-50, and cited references.


Chapter 6

Open Flows: Instability and Transit ion

Convection studied in Chapter 3 is the prototype of systems becoming turbulent by steps when progressively driven far from equilibrium upon increasing the temperature gradient. All along this progression, the physical problem is clearly ruled by a mechanical cause, buoyancy-induced advection, counteracted by thermodynamic dissipation processes, viscous friction and thermal diffusion. Turning to open flows, one could thus try to stay within the same framework and start with the no-flow equilibrium situation, then consider weakly out-of-equilibrium laminar regimes mostly controlled by viscous dissipation, and further increase the shear' to observe the transition t o turbulence after some cascade of instabilities. This approach can indeed be followed in some cases (e.g. wakes) but, in most situations, open flows turn out t o be representative of strongly out-of-equilibrium systems in which mechanical evolution largely preempts the relaxation trends of thermodynamic origin. This has been recognized early since the historical tradition gives the first place to the study of inviscid fluids, for which velocity gradients are not rubbed out by viscous friction. A simple illustration of this change of perspective is given by the damped linear oscillator m X + q X + k X = 0: the high-viscosity limit v2 >> 4km, yielding over-damped behavior (stable node at X = X = 0) typical of thermodynamic relaxation, is not a good starting point t o understand weakly damped oscillatory behavior of inertial origin q2 0 , U(y) = U(+) = Uav + $ A U ,

(6.35)

and first discuss the physical instability mechanism following Batchelor's approach [Batchelor (1967), p. 5111, sketched in Figure 6.7. The unperturbed vorticity sheet is represented by a regular distribution of equal intensity vortices in (a). Next, assume a periodic infinitesimal modulation of the intensity in (b) where points equivalent in a translation by one wavelength are labelled with the same letters and higher local vorticity is represented by arrowed circles with larger diameters. Intuitively, the differential effect of vortices at A and C is to move the interface at B and D in the directions indicated in (c). At lowest order the effect of the base flow is to advect vorticity along the flow, t o the left when the interface has been moved downwards and to the right in the opposite case. Typically, at B it is advected from B toward A and at D from D

Fig. 6.7 Mechanism of the KH instability interpreted in terms of vorticity. The frame is moving at the average speed U,, (coordinate z' = z - Uavt) so that the flow profile displays a jump from - $ A U for y < 0 to + $ A U for y > 0. (a) Unperturbed vorticity sheet. (b) Modulation of the vorticity intensity. (c) Sinusoidal deformation of the interface and amplification of the modulation due to the advection of the vorticity by the base flow.

228

Instabilities, Chaos and Twbulence

Fig. 6.8

Interpretation of the KH instability mechanism in terms of pressure.

toward A, thus increasing the intensity of the vortex at A and decreasing that of the vortex at C. In all cases the initial modulation is increased, hence the instability. A second interpretation can be found in terms of pressure, as illustrated in Figure 6.8; consult, e.g. [Tritton (1990), p. 2681: Before the introduction of a perturbation, the trajectories of the fluid particles on both sides of the discontinuity are straight. When the discontinuity surface is undulated, the speed of the fluid is slightly augmented (reduced) in regions with negative curvature above (below) the surface while in regions with opposite curvature the reverse situation holds. A naive use of the Bernoulli theorem expressing the conservation of energy along streamlines for an inviscid fluid in quasi-steady state, p + $pv2 = Cst, implies that when 1 ~ increases, 1 then p decreases. Consequently a pressure difference tends to build between the two sides of the discontinuity surface, that pushes it further away: the instability develops. All along this discussion, the velocity ratio e = (VZ- Ul)/(VZ V l ) does not show up since the mechanism is analyzed in a frame moving at the average speed U,, = ~ ( U ZU I ) . The physical conditions for the formation of the mixing layer are thus in some sense evacuated. In practice, the trailing edge of the splitting plate has a fixed position in the laboratory frame and, accordingly, a spatial stability analysis in the sense of 56.2.4 below should be developed. Intuitively, when downstream transport is dominant, i.e. Q > 1, i.e. Uz 21 -U1, the situation considered in the previous discussion of the mechanism prevails, the instability

+

+

6. Open Flows: Instability and "Tansition

229

should develop on the spot, and thus be absolute. The flow is then a genuine self-sustained oscillator. The precise value of e at which the behavior change takes place depends on the base profile. For the simplest continuous linear-by-part approximation, (Exercise 6.4.4, second item) it happens at e = 1 and for the smooth hyperbolic tangent profile at e -" 1.3, consult Huerre and Rossi in [Godrkche and Manneville (1998), p. 169ff] for details and original references. 6.2.2.3

T h e Rayleigh equation with linear-by-part velocity profiles

The two previous "physical" approaches rely on a knowledge of basic fluid mechanics properties of inviscid flows relative t o the transport of energy and vorticity that are not necessarily part of the reader's supposed background. It is accordingly valuable to choose a more analytical viewpoint on the KH instability, with just the assumption that it is legitimate to replace the actual base profile by a linear-by-part approximation, splitting the crossstream interval [ymin,ymax] into subintervals such that W ( y ) = &)

+/Wy

for

y E [yj,yj+l].

The second derivative of such profiles with respect to y is identically zero, except at discontinuity points where it is not defined. The Rayleigh equation then simply reads: (6.36) Over each subinterval, the solution of (6.36) is given by: $(j)

=

C ~ 2e x)p ( f k o ) .

(6.37)

(*I In order to find the full solution, it remains to express boundary conditions (6.33) at ymin, ymaxand to match the different partial solutions (6.37) at the discontinuity points of U and/or U'. The matching conditions are obtained from the continuity of pressure in (6.24)

and the continuity of the cross-stream velocity component v at the surface of discontinuity. This second condition derives from the definition of v as the cross-stream velocity at a point of the interface between domains ( j )


230

+

and ( j l),with coordinate Y , hence It reads:

=

&Y (Exercise 6.4.4, first item). (6.39)

Assuming that the interval [ymin,ymax] has been cut into n pieces, 2 n constants (A?), ( j = 1,.. . ,n ) have to be determined. One is left with two boundary conditions and 2 ( n- 1)matching conditions, which indeed makes a system of 2 n homogeneous linear equations. The dispersion relation is obtained from the condition that this system has nontrivial solutions. Let us analyze the KH instability along these lines using the approximate, discontinuous base flow profile (6.35). Solutions to the Rayleigh equation (6.36) are taken in the form (6.37). Forbidding divergence when y + f o o implies

while the matching conditions (6.38, 6.39) at y = 0 read

so that we are left with a homogeneous linear system of two equations for two unknowns. The corresponding compatibility condition:

(U(-) - c)

2

+ (U(+) - c)

2

= 0’

is solved to yield: c=U,,fi$AU,

The instability of the flow directly derives from ci = *!jAU # 0. Since there is no specific scale in this problem (unbounded medium, infinitely thin vorticity sheet) the instability condition is independent of the wavevector k and all modes are unstable. Their growth rates u = kci, are larger and larger as k increases. Things are different for a confined layer -ca < Ymin < 0 < Ymax < +oo, as can be seen by solving Exercise 6.4.4 where several other linear-by-part profiles are also considered.

6. Open Flows: Instability and Transition

6.2.3

231

Viscous flows

6.2.3.1 Instability and viscous dissipation For the KH instability, the destabilizing process is of mechanical origin and viscosity plays a “normal” stabilizing role. But the Rayleigh theorem shows that some flows do not have unstable modes at the inviscid limit. In order to learn what happens to them, we have to return to the Orr-Sommerfeld equation (6.30). The problem is difficult from an analytical point of view owing t o the singular character of the limit R -+ 00. As a matter of fact, whereas the absence of inflection point ys such that U“(ys) = 0 forbids the existence of unstable inviscid modes, neutral modes may exist, for which the Rayleigh problem is singular: at the critical level yc, the phase speed cr is equal t o the speed U(yc) of the flow while U”(y,) # 0. These neutral modes are the best candidates for becoming unstable when perturbed by viscous effects. Here we restrict ourselves to a qualitative presentation of the results, often obtained from a numerical solution of the Orr-Sommerfeld equation, and refer to e.g. [Drazin and Reid (1981)] for the relevant theory. Let us first indicate why viscous dissipation could play a role in the destabilization of a mechanically stable flow. This hint is obtained by adopting a global point of view and considering the evolution of the kinetic energy contained in a specific two-dimensional mode with wavevector k = 27r/X and phase speed c,. Let us compute ( 6 . 1 7 ) ~ ~( 6 . 1 8 ) ~ ~ :

+

uatu + vatv+ uvut + u(vaZv+ U a Z U ) =

+

- ~ & p - d Y p R-’[U(&,

+ aYy)u+ ~ ( 8+. L~$ y ) ~ ] ,

(6.40)

and determine the average kinetic energy (per unit length in the spanwise direction) contained in a domain of length X and height (yma, -ymin), in the frame moving a t speed c, labelled by coordinate 5 = 2 - c,t. The quantity of interest is:

The second term on the 1.h.s. cancels upon streamwise integration by virtue

232


of the periodicity in Z. On the right hand side we have - u a x p - w a y p = -ax(up) -a,(~p)+p(a,u+d,v). The two first contributions cancel upon integration, either owing to streamwise periodicity or t o cross-stream boundary conditions at ymin and ymax,and the last one from the continuity condition a X u + ayw= 0. Finally we get:

&Kpert= -(U’UW) - R-’(((&U)~ + ( 8 , ~+) ~

+ (a,~)’)),

(6.41)

where the viscous terms (in R-l) have been rearranged through integrations by parts exploiting the boundary conditions. This is the specific form taken by the Reynolds-Orr equation governing the evolution of the kinetic energy contained in the perturbation in the linear regime.3 Equation (6.41) symbolically reads: ZKpert d

=P -D

(6.42)

>

where P is a production term and D, that contains all the terms in R-I, accounts for viscous dissipation. Term D is manifestly always positive and works so as to decrease the energy contained in the perturbation. The flow can thus be unstable only if the production term P is positive and sufficiently large. Let us consider perturbations in the form

q{-

{u,w} = 2 U(Y),fi(Y)} e x P [ w x - 4 1+ C.C.) = $ ( { i i ( y ) , ~ ( y ) }exp(ik2) c.c.) exp(kQt),

+

with growth rate kci. It can then be checked that every term in (6.42) varies as exp(2kc;t) that can be factored out, leaving us with a discussion of the sign of

P cc -

/

U‘(y)[ii(y)V*(y)

+ c.c.] dy ,

which is not determined in advance. On the contrary, this sign strongly depends on the shape of u and w , solutions t o the full Orr-Sommerfeld equation (6.30). The asymptotic analysis in the limit ‘ R -+ 00’ points out the role of the variation of the stream function close t o the critical

4

3This equation is valid without modification in the nonlinear regime because the advection term conserves the energy. Terms of nonlinear origin and dropped upon linearization on the 1.h.s. read u2&u u ~ t 3 ~ uvu&v v 2 $ u , further rewritten as i & ( u ( u 2 v 2 ) ) $t3q(v(u2 v ’ ) ) using the continuity condition. All these terms disappear in the averaging process. Even more, the result is not restricted to the twodimensional case but also holds for three-dimensional perturbations.

+

+

+

+

+

+

6 . Open Flows: Instability and Transition

233

level yc, and shows the existence of specific unstable modes called TollmienSchlichting (TS) waves. Rather then starting a long and delicate discussion that would allow one to compute the behavior of the marginal stability curve in this limit (for an introduction, consult Huerre and Rossi in [Godr&che and Manneville (1998)l) we just present the main results obtained in part analytically and in part numerically.

6.2.3.2 Results By contrast with convection-like instabilities (see Fig. 3.3, p. 81), marginal stability curves are conventionally presented in the plane ( R ,k) with R along the horizontal axis and k vertically. Whatever the origin of the instability, mechanical or viscous, they take the shape of loops with different aspects in the limit ‘ R + 00’, as shown in Figure 6.9. 0 When the flow is mechanically unstable (velocity profile with an inflection point and a vorticity maximum at the inflection point), the threshold is “low” and a whole band of wavevectors remains unstable as R + 00, see Figure 6.9 (left). Viscosity then plays its usual stabilizing role.

When the base profile is mechanically stable (no inflection point), the threshold is “high” and the marginal stability curve has the shape of a hairpin that pinches as R + 00, see Figure 6.9 (right). Unstable modes are the TS waves resulting from the viscous mechanism alluded t o above. 0

k

Fig. 6.9 Marginal stability curves typical of plane shear flows. In each case, the unstable domain corresponds to the interior of the loop. Left: Mechanically unstable flows; low threshold and finite-width band of unstable wavevectors when R --t 00. Right: Mechanically stable flows (no inflection point); high thresholds and narrowing band of unstable wavevector when R --t M.

Instabilities, Chaos and -1Lrbdencecce

234

Plane Poiseuille flow in a channel is typical of flows without inflection point. Taking the maximum speed Urn,, on the centerline of the flow as velocity unit and the half-gap h/2 = (ym,, - ymin)/2 as length unit. the dimensionless base velocity profile is thus U = 1 - y2 over the interval [-1,+1] and the Reynolds number is defined by R = Um,,h/2v. The marginal stability curve looks like that in Figure 6.9 (right), with Rc N 5772 >> 1 and k, N 1.02 (critical wavelength A, = (h/2)(27r/1.02) N 3h).4 The phase speed of waves at threshold is c,,~N 0.264 ( i e . about 1/4 of Urn,,). To these numerical results one can add the asymptotic behavior of the marginal stability branches obtained analytically (see e.g. [Drazin and Reid (1981)l for a detailed presentation and results summarized in their Fig. 4.11, p. 190):

6.2.4

Upper branch

+ R1I3 N 8.44 (k2)-11/3,

Lower branch

+ R 1 / 3N 5.96 ( k 2 ) - 7 / 3 ,

cr N &k2 c,

N 0.611k2.

Instability and downstream transport

6.2.4.1 Theory us. experiments Let us come back t o the dispersion relation (6.23). The Squire theorem tells us that the wavevector kh has a single relevant (streamwise) component noted k, so that we can write

L(-iw,ik;R) = 0 .

(6.43)

The presentation of results in the plane (R, k ) corresponds t o a temporal reading of this relation, further solved by assuming k E R and w complex, w = w, +iwi, so that wi is the temporal growth rate (from the normal mode assumption: exp[i(kx - wt)] = exp(wit) exp[i(kx - wrt)]). This approach was satisfactory in the case of convection that develops in an enclosure and emerges from the background noise, in the absence of any forcing (absolute instability). It is no longer appropriate for typical open flow situations like the one sketched in Figure 6.10, where downstream transport manifestly plays a dominant role (convective i n ~ t a b i l i t y ) In . ~ such a setup, a localized forcing, periodic with angular frequency wf is applied at 4S.A. Orszag, “Accurate solution of the Orr-Sommerfeld stability equation,” J . Fluid Mech. 50 (1971) 689-703. 5 The terms absolute and convective are the accepted ones. We avoided to use the word ‘convective’ all along Chapter 3 dealing with ‘convection’ which is ‘absolute’ in the absence of (sufficiently strong) through-flow.


detector

235

anemometer

5 0

>

/

wave absorber

’

Fig. 6.10 Experimental configuration used in the film “Flow Instabilities” [MolloChristensen (1972)]: A wind is blown over a water channel, produced by a vacuum cleaner downstream, and settled by the honeycomb upstream; its speed is measured by an anemometer. A wave generator hits the surface close to the entrance, producing a localized perturbation with given frequency. The height of the waves is measured by an appropriate gauge at a fixed distance from the wave generator.

a given point in the Aow and the evolution of the perturbation is recorded at downstream stations as a function of the forcing characteristics and the Reynolds number. From the records, a downstream spatial amplification rate can be defined, negative when the perturbation is damped, positive otherwise. The different behaviors observed, sketched in Figure 6.11, can be analyzed in terms of plane waves with angular frequency wf, exp[i(kx - w f t ) ] , modulated by a slowly decaying or growing envelope, exp(px), with p small, hence exp[i(bz - wft)]e x p ( p ) . The two exponential factors can be recombined as exp[i(k - zp)x - wft] to form a complex wavevector with real part k and imaginary part -p. We are thus lead to a spatial reading of the dispersion relation (6.43) solved with w = wf real and k complex. This simplistic presentation is not free from criticism. We have indeed avoided evoking what happens upstream of the forcing point and represent it in Figure 6.11. While -ki > 0 corresponds t o an amplification for x > 0 (which conventionally denotes the downstream direction), a mode with -ki < 0 is amplified upstream and the picture is reversed. The nonequivalence of upstream and downstream directions due t o advection has delicate analytical implications that we still mostly skip, asking the reader to turn to specialized references.6 ‘ e . 9 . the review article by P. Huerre, “Open shear flow instabilities,” in [Batchelor et al. (ZOOO)] and references quoted therein.

236


Ih 0

X

-5

0

-5

Fig. 6.11 The spatial amplification rate of the waves is extracted from the response of the system to a localized periodic excitation. The marginal value Rm(w) of the control parameter separates the domain of damped waves R < Rm(w) from that of amplified waves R > Rm(w).

Without entering mathematical intricacies, following Gaster’ we can however relate the main characteristics of the spatial (S) problem to those of the temporal (T) problem. Let us consider the dispersion relation (6.43) and assume that it has been analytically solved for w as a function of k , both complex, at given R:

w = w(k; R).

(6.44)

The expansion of the dispersion relation around some arbitrary point ( k ( O ) w(’)’> , solution to (6.43) reads (6.45) (6.46)

A nearly neutral T-mode is characterized by:

and the corresponding angular frequency

7M. Gaster, “A note on the relation between temporally increasing and spatiallyincreasing disturbances in hydrodynamic stability,” J. Fluid Mech. 14 (1962) 222-224.

6. Open Flows: Instability and B a n s a t i o n

237

We are looking for the characteristics of an S-mode (k!”, k/”) with k!’) = kiT) and angular frequency wiS) with w:’) = 0 by definition. The T and S modes fulfill the dispersion relation (6.44) and are both nearly neutral, which means in particular ki(’) = ki # 0 with (ki(’)I 0 for some k . 0 It is linearly stable when wi < 0 for all k . These situations are depicted in Figure 6.13 which displays perturbation profile as a function of space at several successive times. An elementary illustration of the use of the criterion is considered in Exercise 6.4.5. The preceding discussion is typically local in that is applies to a quasiparallel flow provided that its 2 dependence can be neglected. In this perspective, when the flow develops downstream, the intensity of the instability mechanism varies in space so that its absolute/convective character may change downstream. In that case only a global analysis can help us understand the stability properties of the flow, which is another full story in itself. Finally, in all that precedes departures from the base flow were tacitly assumed to be wave-like and infinitesimal, making the linear approach legitimate (localized finite-size perturbations were discarded). Besides the identification of important instability mechanisms, a thorough understanding of the transition to turbulence in developing flows has to face difficulties arising from a combination of alternatives: absolute vs. convective, local us. global, and linear us. nonlinear.


t=to+2At

“1, ,’

241

, ,, ,,” X

0;

t=t X

Fig. 6.13 Space-time representation of the growth/decay of a perturbation introduced at x = 0 and t = 0. Top-left: The two edges of the wavepacket go in opposite directions and the perturbation grows on the spot, the instability is absolute. Top-right: The perturbation grows but is evacuated since the two edges of the wavepacket go in the same direction, the instability is convective. Bottom: The perturbation vanishes and the flow is stable.

6.3

6.3.1

Transition to Turbulence

Nonlinear development of instabilities

6.3.1.1 Saturation of the primary mode In its principle, the approach is parallel to that followed in Chapter 3 for natural convection. In order to identify the different steps more clearly let us introduce subscripts, ‘0’ for the base state, ‘1’ for the primary perturbation, ‘2’ for the secondary modes, etc. The linear stability theory of the parallel base flow VOyields a critical primary mode V1 in the form of a two-dimensional wave with wavevector k aligned with the flow direction, phase speed c, and no spanwise flow component. Let Al be the amplitude of this mode. The first question to be answered relates t o the saturation of A1 beyond threshold, which is dealt with by means of an expansion in powers of the amplitude. This 6 la Landau approach has been introduced in the field of open flows separately by Stuart and Watson in 1960. Being a

242

Instabilities, Chaos and n r b d e n c e

stationary cellular instability, convection is accounted for by a supercritical fork bifurcation governed by an equation for a single variable with real coefficients. By contrast, here the wavy behavior of the flow beyond threshold makes it time-oscillatory in the laboratory frame so that a Hopf bifurcation on a two-dimensional center manifold is expected, see Chapter 4, Eq. (4.16), p.128. With s1 = (TI - iwl where (TI is the real growth rate of the primary mode (positive when unstable) and w1 the angular frequency, changing for a rotating amplitude by setting A1 = 21 exp(-iwlt), we get:

+

where the complex coefficient g = gr igi describes the effective interaction of the unstable mode with itself (through the coupling with other, adiabatically eliminated, non-critical modes, see Chapter 4, 54.1). Saturation is controlled by the sign of the real part gr of g. When gr > 0 the bifurcation is supercritical and the wave saturates at an amplitude that smoothly varies with the distance to threshold. In the opposite case, the wave does not saturate at lowest order, one must continue the expansion (see the case of the plane Poiseuille flow discussed later) but there is a risk that no stable branch exists at finite distance from the base solution, as recognized early by Landau (Note 9, p. 92). 6.3.1.2 Secondary instabilities Let us now sketch what should be the next steps and assume that the nonlinear problem for the primary mode has been solved and that saturation at some amplitude A1 has been obtained. The stability analysis is then repeated for the flow V O=~VO AlVl in a frame moving at the speed cnl of the saturated wave (coordinate 2 = z - cnlt). In subscript ‘01’ the ‘0’ indicates that it is the new base flow and the ‘1’ that it arises from the saturation of the primary perturbation. Concrete computations are particularly involved. The standard linearization process leads to a partial differential system with coefficients periodic in 1 with period X = 2.rrf k. For such linear operators, the Floquet theory implies taking secondary normal modes VZin a form that explicitly isolates nontrivial space-time departures from a trivial space dependence in 3 with period A:

+

I

VZ= A ~ e x p [ i ( q , 2 f q , t - - w ~ t ) ] V ~ withVz(Z+X) (Z) = V Z ( Z ) . (6.51) The exponential part accounts for a supplementary modulation introduced


243

=

by the putative secondary instability, with real wavevector q (q5, qz) and complex angular frequency w2 = ~ 2 W , Z ~, ~(temporal instability scheme). The main possible cases of this conceptually well-posed problem can be identified without doing any explicit calculation, just by considering expression (6.51). First it is not difficult to show that the value of qx/k can be restricted to the interval [0,1[ since, setting qs = (c n)k with K E N,one gets:

+

+

exp(iqxz) = exp[27ri(c

+ K)IC?]

= exp(2i7rsk?) exp(2i7r~k?),

where the last exponential factor is periodic in 3 with period 21r/nk, thus also periodic with period 21r/k, so that it can be incorporated into that accounts for the trivial part of the space dependence. Next, possible secondary modes can be classified according to whether (i) qt = 0 or qz # 0, (ii) qx = 0 or qx # 0, and (iii) when qx # 0, whether qx is commensurate or not with k, i.e. c rational or irrational. In particular: When qz = 0 and qx = k/2 (c = 1/2), the secondary mode is still two-dimensional but has a wavelength twice that of the primary mode, this period doubling instability is called pairing. 0 When qt # 0 and qx = 0, the secondary mode is most often called f u n d a m e n t a l , while the term subharmonic is associated to the case qz # 0 and qz = k/2 (see later, Figure 6.18). When qx is not commensurate to k (c irrational), the situation is similar t o that of quasi-periodic temporal systems considered in Chapter 4,$4.2.3, and can be transposed from it. The physical mechanisms involved in secondary instabilities have been studied both theoretically and experimentally. One generally makes a distinction between viscous processes evolving over long time scales (in practice mostly the growth of TS waves) and inertial processes developing over short time scales. In this latter context, the elliptical instability affecting the core of vortical structures is of great importance. For a detailed discussion with references, see e.g. Kerswell“ or Huerre and Rossi in [Godrkche and Manneville (1998), p. 269ffl. Rather than considering the “natural” transition, the comparison of theory with experiments often bears on flows in which the spanwise dependence of the secondary mode is forced at a given qr by some experimental trick, e.g. by grooving the wall in a boundary layer experiment, the splitting plate for a mixing layer, the lips of the slit for a plane jet, etc.

v2

‘OR.R. Kerswell, “Elliptical instability,” Ann. Rev. Fluid Mech. 34 (2002) 83-113.

244

Instabilities, Chaos and l b b u l e n c e

The tertiary instability is in general hardly accessible t o theory and small scale turbulence usually appears soon after secondary modes set in. The last step of the transition to turbulence is often interpreted as being due to inflection points in the velocity profiles resulting from the superposition of the primary and secondary modes to the base flow, thus promoting KH instabilities at the origin of the smaller scales. On the whole, the situation is particularly involved since space and time aspects of the development of transitional flows are intermingled in a complicated way owing to downstream transport and evolution. In the rest of this chapter we present salient features of the transition to turbulence a t a phenomenological level resting on the dichotomy between inviscidly unstable and stable flows and the supercritical or subcritical character of the bifurcation. Rather than trying to be exhaustive, we focus on the most typical situations in each case.

6.3.2

Inviscidly unstable flows

6.3.2.1 General features Two-dimensional base flows that are unstable at the inviscid limit display inflection points in their velocity profile (Rayleigh and Fjortoft theorems). These are mixing layers [Fig. 6.6), jets and wakes (Fig. 6.2). The transition to turbulence is characterized by primary instabilities that set in a t low Reynolds numbers in the form of two-dimensional spanwise structures as expected from the linear theory. These structures then gently saturate beyond threshold to form developed vortices. Secondary modes are most often streamwise modulations (qz = 0) leading t o quasiperiodic behavior and lockings, especially subharmonic locking that manifests itself as vortex pairing (qZ = k/2). In the case of mixing layers, successive pairings may take place before a secondary instability can introduce some three-dimensional dependence. Small scale turbulence enters soon after the introduction of three-dimensional modes usually arising from the elliptical instability previously mentioned (Note 10). The plane jet, understood as two side-by-side mixing layers close t o the outlet and merging further downstream, behaves in the same way. Another general feature of this transition scenario is the persistence of coherent structures in the post-transitional regime. These are large vortices with lifetimes long when compared to their turn-over times, resembling KH vortices but now developing over a turbulent mean %ow profile.

6. Open Flows: Instability and “ransition

6.3.2.2

245

Wakes

In the following, we focus on the case of the wake of a cylinder, a quantitatively much studied system which combines specificities of flows displaying inflection points in the base flow t o features arising from the absolute character of the instability which is responsible for its self-oscillatory nature. Here we borrow mainly from the review given by Williamson.” The distinctive feature of the wake behind a bluff body is the occurrence of a sharp Hopf bifurcation that marks the emergence of very regular periodic vortex shedding forming a pattern known as the Ka’rma’n vortex street after von K&rm&n(1911-12)l’ and already depicted in Figure 1.6, p. 16. The origin of this behavior can be understood by noticing that, at flow rates corresponding to the bifurcation point, a steady recirculation develops just behind the obstacle so that the velocity profile displays a region where downstream transport is compensated. By contrast, away from it, the flow has been smoothed by viscous effects and is closer to the uniform flow blowing from infinity around the body (Fig. 6.14). Accordingly, there is a region of the flow at the rear of the obstacle where the instability linked to the presence of inflection points can develop on the spot (hence ‘absolute’), producing the oscillations. This situation is reminiscent of what happens in closed systems where confinement effects select specific modes and nonlinearities can be studied in the framework of dynamical systems theory (Chapter 3, $3.2.3 and Chapter 4, $4.2). Let us consider a cylinder placed perpendicular to a uniform flow13 with speed U,. This experimental configuration is characterized by two dimensionless parameters. The first one is the Reynolds number:

R = U,d/v, where d is the diameter of the cylinder and Y the kinematic viscosity of the fluid. The second parameter is the aspect-ratio r = L/d where L is the llC.H.K. Williamson, “Vortex dynamics in the cylinder wake,” Annu. Rev. Fluid Mech. 28 (1996) 477-539. 12T.von KBrmln, “Uber den Mechanismus des Widerstandes, den ein bewegter Korper in ein Fliissikeit erzeugt,” Nachr. Ges. Wiss. Gottingen, Math. Phys. Klasse (1911) 509-517, (1912) 547-556. Previous experimental observations by H. BBnard (1908) are reported in Notice sur les titres et travaux scientifiques de M. Henri Binard (GauthierVillars, Paris, 1926). French chauvinism often associates the two names with the wake instability. 131n one of the last sequences of the film “Flow instabilities” [Mollo-Christensen (1972)], a cylinder is dragged through a viscous fluid at rest. In that case U, is of course the speed of the body.

246


recirculation region

4

Fig. 6.14 ‘Absolute/convective’ transition in the wake of a cylinder (and, more generally, of a blunt body).

length of the cylinder. Here we assume L >> d (or I? + m), so that it seems legitimate to neglect end effects and to start with a two-dimensional base flow. Accordingly, we are left with R as the only parameter. The flow structure can be understood from visualizations of the velocity field, while the different regimes are most conveniently identified through the variation of global observables. A first one is the oscillation frequency f, or in dimensionless form, the Strouhal number

A second classical measure is the drag, i.e. the force FD necessary t o maintain the cylinder at its position, or rather the dimensionless drag coefficient CD = FD/fpU&,d. Another related quantity is the base pressure, the pressure difference between a point at 180 degrees from the upstream stagnation point and a reference pressure, usually the static pressure at infinity. Here we will just discuss the early stages of the transition using the Strouhal number. The transition to turbulence behind a cylinder is described at many places besides Williamson’s review (Note ll), e.g. in [Tritton (1990)l. Beautiful pictures of the different laminar flow regimes by Taneda can be found in [van Dyke (1982)]. Creeping flow with flow lines smoothly sticking t o the cylinder exists up to R 5-6, beyond which a steady symmetrical bubble of recirculating fluid sets in. This time independent situation persists up to R = R, N 48.5 beyond which the wake oscillates, periodically emitting vortices parallel to the cylinder axis as illustrated in Figure 6.15.

-


I

parallel shedding

247

oblique shedding

Fig. 6.15 Top: Rerspective view of the shedding of parallel vortices, after one of Taneda’s pictures in [van Dyke (1982)]. Bottom: Parallel vs. oblique shedding.

At threshold, the Strouhal number is of order 0.12, so that the streamwise wavelength A0 = Urn/f is of order 8d. Beyond threshold, the oscillation frequency increases regularly with the Reynolds number as shown in Figure 6.16. In fact the frequency selection problem is a difficult one, both experimentally and theoretically. On the experimental side, it took some time before it was recognized that the dispersion of the observed frequencies was due t o the occurrence of oblique shedding, i.e. shedding with the axis of the vortices at an angle 0 with the cylinder axis. Assuming that the frequency of parallel shedding is fo, a simple geometrical construction (Fig. 6.15, bottom-right) shows that the frequency of oblique shedding (and with it, the Strouhal number S) is reduced by a factor cos0: let Ae be the wavelength of the oblique vortex street as measured perpendicularly to the vortex axis and suppose that the instability mechanism makes it identical t o that of parallel shedding, then the wavelength measured along the flow direction is longer by a factor 1/ cos0; to get the result one just needs to convert wavelengths into oscillation periods using the speed of the flow U,. Rescaling frequencies accordingly yields the master curve displayed in Figure 6.16 (left). The theoretical understanding of the neighborhood of the threshold is not as obvious as it may seen. At first, the wake can be thought of as a

248


Is

s

o,o/

R 40

60

80

100

120

140

160

160

/ 100

R 200

300

404

Fig. 6.16 Variation of the Strouhal number S (dimensionless frequency of shedding) as a function of the Reynolds number R . Left: Two-dimensional shedding regime corrected for the effects of oblique shedding. Right: Transition to three-dimensionality. Modes modes A* and B’ have the same structure than modes A and B but display dislocations; dashed lines correspond to unstable regimes; The transition ‘2D ts mode A*’ is hysteretic; the “natural” route is: 2d -+ mode A* --t mode B. After Williamson, note 11.

collection of Hopf oscillators, distributed all along the cylinder and coupled to their neighbors. An appropriate local model would then be (4.16), p. 128 which, once unfolded in the spanwise direction z , would yield a complex Ginzburg-Landau equation (5.36),p. 198. The dynamics of the wake close t o the threshold is in fact reasonably well described by this equation whose coefficients can be fitted against experiment~.’~ Chevron patterns observed in case of oblique shedding, finite-size effects, or the presence of dislocations in the vortex system (viewed as phase defects for the complex amplitude), can be understood within this phenomenological approach. But this cannot be the whole story since the approach implicitly assumes that the streamwise )(x) .( structure of the perturbation is frozen, which this is not the case. While the amplitude of the maximum of the wake velocity fluctuations indeed increases as the square root of the distance t o threshold as expected for a Hopf bifurcation, the position of the maximum shifts upstream closer to the cylinder. Progress has been made recently regarding the variation of the streamwise shape of the vortex amplitude and the frequency selection problem by combining convective/absolute, global, and nonlinear issues.15 14M. Provansal, C. Mathis, L. Boyer, “BCnard-von Ktirman instability: transient and forced regimes,” J. Fluid Mech. 182 (1987) 1-22. Th. Leweke, M. Provansal, “The flow behind rings: bluff body wakes without end effects,” J. Fluid Mech. 288 (1995) 265-310. ”See, e.9. B. Pier and P. Huerre, “Nonlinear self-sustained structures and fronts in spatially developing wake flows,” J . Fluid Mech. 435 (2001) 145-174, B. Pier, “On the frequency selection of finite-amplitude vortex shedding in the cylinder wake,” J. Fluid Mech. 458 (2002) 407-415, and references cited.

6 . O p e n Flows: Instability a n d T r a n s i t i o n

249

Beyond R x 150, the wake enters a transition regime where the regular vortex street may be disrupted by several different secondary instability modes, all involving spanwise modulation. The picture is further complicated by the presence of vortex dislocations. Two main modes have been identified,16 first experimentally, next theoretically through linear stability a n a 1 y ~ e s . lThe ~ first one, called ‘mode A’, arises as a deformation of the primary vortices with wavelength of the order of 0.6-0.8Xo, where A0 is the primary wavelength at the corresponding value of R. Since the Strouhal number is then about 0.2, the wavelength of mode A is of order 3-4d. By contrast the second one, called ‘mode B’, has a much shorter wavelength of order 0.2X0, i.e. of the order of d. The physical origin of these modes cannot be explained in terms of the elementary KH mechanism discussed previously but rather relates to the elliptical instability (56.3.1.2 and Note 10). The transition regime extends up to R x 260. Beyond this point the intensity of fine structures increases and the nature of the instabilities involved in the primary vortex shedding mechanism changes gradually. Rather than deriving from the global mode attached t o the recirculation bubble, as was the case at lower R, these instabilities have to do first with shear layer that forms downstream the detachment point and, at the highest Reynolds numbers, with the boundary layer that develops along the cylinder itself.

6.3.3

Inviscidly stable flows

6.3.3.1 Plane Poiseuille flow This prototype of Aows without inflection point has been much studied both theoretically and experimentally. The threshold of the instability against T W waves is R, E 5772 (see p. 234). As shown by Herbert’’ the bifurcation is subcritical with unstable nonlinear steady states appearing below threshold, but the bifurcated branch turns back a t R,1 21 2900 (Fig. 6.17). The next step should be the linear stability analysis of two-dimensional saturated TS waves against three-dimensional infinitesimal perturbations, as suggested in 56.2.1. One would thus expect transversally modulated 16Ch. Williamson, “Three-dimensional wake transition behind a cylinder,” J. Fluid Mech. (1995). “D. Barkley, R.D. Henderson, “Three-dimensional Floquet stability analysis of the wake of a circular cylinder,” J. Fluid Mech. 322 (1996) 215-241. “T. Herbert, “Secondary instability of plane channel flows to subharmonic threedimensional disturbances,” Phys. Fluids 26 (1983) 871-874.

250


A

1.

II1 ,

,,/unstable

branch

I

Fig. 6.17 Bifurcation diagram of two-dimensional TW waves in plane Poiseuille flow. After Herbert, Note 18.

waves later decaying into small scale turbulence. Several difficulties hinder the observation of this scenario. First, the primary instability is convective so that detecting its presence in a finite length channel strongly depends on the level of background noise. Second, the bifurcation is subcritical so that working with a “clean” base flow inevitably leads to missing the stable bifurcated branch at finite distance from it, whereas triggering appropriate waves, e.g. using vibrating ribbons, is not so easy. By contrast, a moderate high level of residual turbulence brings about a different scenario (natural transition) involving localized finite-size perturbations, called turbulent spots, i. e. limited patches of small-scale turbulent flow traveling amid laminar Poiseuille flow, growing in size as they process downstream.’’ These turbulent spots are transient and rapidly decay into laminar flow below R, M 1000, but are sustained beyond this value, which is much lower than R, or even R,1. The value R, plays the role of a global stability threshold, hence the notation (see p. 9 for a refresher). Except for the existence of a linear threshold and a corresponding nonlinear branch, the situation is similar to that taking place in the plane Couette flow to be examined later in $6.3.4.3.

6.3.3.2 Boundary layer flows The main drawback of the Poiseuille flow as a test case for continuous transition scenarios in open flows stems from its strongly subcritical character. leD.R. Carlson, S.E. Widnall, M.F. Peeters, “A flow visualization of transition in plane Poiseuille flow,” J. Fluid Mech. 121 (1982) 487-505.


251

Boundary layer flows may seem more favorable from this point of view and, being ubiquitous, have even more practical interest. The Blasius velocity profile is another example of flow without internal inflection point. As we have seen, the thickness 6 of the laminar boundary layer increases as the square root of the distance to the leading edge. Theory predicts the emergence of TS waves a t R = R, = 519.4 with k, 0.303 and cr,=N 0.397 (all scalings based on 6). The bifurcation seems supercritical or nearly so, but the interpretation is made delicate by the downstream dependence of the flow. Anyway, linear stability has been tested at low residual turbulence levels in vibrating ribbon experiments and the agreement between predictions and observations is satisfactory once corrections due to the slight divergence of the flow are taken into account. Several tricks have been used to attempt a t controlling the spanwise wavevector qt of the secondary modes in experiments (regularly spaced grooves or adhesive tapes aligned along the flow direction). In practice the most dangerous secondary modes either do not add streamwise modulation (q2 = 0, < = 0, where c was introduced in 56.3.1.2) or else introduce a spatial subharmonic (qz = k/2, that is q = 1/2). In the latter case, resonant interactions between a mode triad formed with the primary mode (k,0) and two oblique waves (q2 = k / 2 , &qz) may be expected from the quadratic coupling through the advection term v Vv (Figure 6.18, top). The velocity field resulting from the superposition of the base flow and the secondary modes displays streamwise bands with alternatively increased and reduced speed (“peak-valley” alternation) that, upon amplification, take the shape of so-called h-vortices, see Fig. 6.18, bottom. A step-by-step transition is thus expected. Laminar flow first becomes unstable against two-dimensional TS waves, the amplitude of which slowly increases over a viscous time-scale. TS-waves then experience growth of secondary modes introducing three-dimensionality over a much shorter inertial time-scale, and a final breakdown into small scale turbulence. In fact the kind of secondary mode that develops the fastest depends on the amplitude A1 reached by the TS waves, which in turn is a function of the level of residual turbulence (through the exponential amplification of the primary mode over a limited time, converted into a distance by the downstream flow). For a detailed presentation consult the reviews by Herbert and/or Kachanov,” and also [Schmid and Henningson (2001)j.

=

“T. Herbert, “Secondary instabilities of boundary layers,” Ann. Rev. Fluid Mech. 20 (1988) 487-526; Y.S. Kachanov, “Physical mechanisms of laminar-boundary-layer transition,” Ann. Rev. Fluid Mech. 26 (1994) 411-482.

252

Instabilities, C h a o s and Turbulence

Fig. 6.18 Top: Resonance conditions for secondary modes. Bottom: (a) Aligned hvortices, qz = 0, ‘K’ mode. (b) Staggered h-vortices, qz = k / 2 , ‘H’ mode.

Figure 6.19 adapted from [Schlichting (1979)l displays a global schematic image of these different steps. At the lowest residual turbulence levels, the amplitude of the TS waves staying below 0.2-0.3% of the speed at infinity, relaminarization is observed owing to the shape of the marginal stability curve, as already mentioned). When their amplitude is between 0.3% and 0.6%, the fastest emerging secondary mode is subharmonic, which produces staggered A-vortices, this is ‘mode H’ after Herbert’s name, Note 18, p. 249. Finally, when their amplitude is larger than 0.6%, the prevalent secondary mode is with qz = 0 and produces aligned A-vortices as early observed by Klebanoff and coworkers.’’ A-vortices then breakdown into spikes that serve as germs for turbulent spots developing first in the cross-flow direction y and then laterally in the z-z plane while being advected downstream. Merging of the turbulent patches yields a fully developed turbulent boundary layer. Large streamwise rolls called wall streaks, with finite lifetime and well-defined wavelength can be identified within the turbulent boundary layer. They experience intermittent breakdown and regeneration. This phenomenon called bursting 21P.S. Klebanoff, K.D. Tidstrom, L.M. Sargent, “The three-dimensional nature of boundary layer transition,” J. Fluid Mech. 12 (1962) 1-34.

6. Open Flows: Instability and fiansition

stability

primary instability TS-waves (2d)

secondary instability and 3d transition

253

developed hubulence

Fig. 6.19 Side view and top view of a ‘K mode’ transitional boundary layer. After [Schlichting (1979)l.

has been studied within the framework of chaotic dynamical systems using tools similar to those presented at the end of chapter 4, $4.4.22 These different steps are difficult to identify in the absence of control of secondary modes (natural transition) and turbulent spots often appear without apparent precursors. Such spots can also be introduced by perturbing the laminar boundary layer locally and sufficiently strongIy. Of course, when the flow is very clean, one must not conclude that it never become turbulent, but that, when it does, the other scenario involving spots develops. A kind of competition between the standard sequences sketched above and a bypass transition not relying on secondary instability modes can thus be observed in the general case. 6.3.4

Turbulent spots and i n t e m i t t e n c y

6.3.4.1 C o n t e x t In several instances we have mentioned the fact that turbulent spots, i.e. bounded regions filled with strongly turbulent flow, could appear, move, and expand within laminar flow. This phenomenon can be understood by noticing that that solutions to the Navier-Stokes equations may not all be derived continuously from the thermodynamic branch by increasing the ”N. Aubry, Ph. Holmes, J.L. Lumley, E. Stone, “The dynamics of coherent structures in the wall region of a turbulent boundary layer,” J . Fluid Mech. 192 (1988) 115-173.

254


Fig. 6.20 (a, b) Super/sub-critical local bifurcation. The distance to the basic state is measured by the amplitude A of the primary instability mode beyond the corresponding threshold R,. (c, d) Global bifurcation toward a branch of nontrivial states existing beyond R > Rnt, disconnected from the basic state, A is the measure of the distance to that state.

Reynolds number after supercritical bifurcations, and that there may exist other fully nonlinear solutions belonging to other disconnected branches. Figure 6.20 illustrates this situation in a highly simplified way: The bifurcation diagrams (a) and (b) correspond t o local bifurcations and have the same precise technical meaning as in Chapter 4: variable A on the vertical axis is the amplitude of the bifurcated state. By contrast, the nature of the states on the nontrivial branch in diagrams (c) and (d), resulting from global bifurcations, is left unspecified for the moment. Quantity A, which serves as an order parameter, is simply some idealized statistical measure of the distance to the base flow, hiding a multiplicity of (possibly irregular) flow configurations. From these last two sub-figures, one will just retain (i) the possible existence of nontrivial states for R > Rnt (‘nt’ for ‘nontrivial’) and (ii) the fact that depending on cases, the local bifurcation can take place at values of R lower or higher than Rnt. Diagram (d) represents a case with R, > Rnl and Rnl > Rnt, i.e. a subcritical, primary, local bifurcation and nontrivial states at even lower values of R, reminiscent of our description of the plane Poiseuille flow transition. Diagram (c) with a supercritical primary bifurcation occurring first R, < Rnt, and a nontrivial branch soon becoming relevant, would rather model the case of the Blasius boundary layer, though the exact situation is difficult to appreciate due t o the downstream development of the instability. At any rate, the suggestion is made explicit that the globally supercritical cascade toward turbulence can be preempted by other mechanisms involved in the so-called bypass transition (that can thus be consistently termed globally subcritical). In terms of dynamical systems theory, the principal


255

b

Fig. 6.21 Transition to turbulence by growth of turbulent plugs/slugs in a cylindrical pipe. The growth, propagation, and merging of the slugs is suggested by sketches of snapshots taken at regularly spaced successive times.

difficulty is then to determine the boundary of the attraction basin of the nontrivial solution, that is to say not only the most dangerous directions in phase space, from the base state or related “trivial” nonlinear solutions, but also the amplitude of the perturbations that actually trigger the transition. Solving exercise 4.6.6 in Chapter 4 may contribute to the understanding of that difficulty in an exceedingly simplified case, the precise connection of that exercise with the present problem is discussed in the article from which it derives, Note 27, p. 174. The case of plane Couette flow is, from this point of view, the most dramatic since the base flow is linearly stable for all R as shown by ROma no^,^^ so that the branch that would correspond t o the local bifurcation is pushed a t infinity (in R). This circumstance is shared by the Hagen-Poiseuille flow, the laminar flow in a cylindrical pipe, which is not a plane flow but is mentioned here owing to the historical role of Reynolds’ experiment (1883) regarding the definition of the Reynolds number and the problem of the transition to t ~ r b u l e n c e It . ~ indeed ~ turns out that the parabolic velocity profile (6.5) is also linearly stable for all R, though turbulent flow can be observed for sufficiently large R. The transition occurs without intermediate steps through the formation of turbulent plugs equivalent t o the turbulent spots observed in plane flows. As suggested in Figure 6.21, a turbulent plug develops close to the tube wall, then invades the section and grows in the streamwise di23V.A. Romanov, “Stability of plane parallel Couette flow” (english transl.) Funktsional’nyi Analiz i Ego Prilozhaniya 7 (1973) 62-73. 24The experiment is repeated in the film nrbulence [Stewart (1972)]. There, the Reynolds number is modified by mixing water and glycerol in order to vary the viscosity, rather than by changing the flow rate.


256

rection to become a turbulent slug. These slugs grow in length and thus persist indefinitely, i.e. till the end of the pipe, provided R is large enough. In practice, the head of a slug moves faster than its tails for R greater than about 2300. The slugs become statistically longer and longer beyond this value and, if their birth probability is large enough, the pipe becomes turbulent along most of its length. The transition is thus highly sensitive to the care brought to the experimental conditions (shape of the entrance of the pipe, roughness of the wall, level of residual turbulence). The generation mechanism of the plug depends on the Reynolds number. Laminar flow can be maintained at R > lo5 in experiments designed with extreme care but usually the natural transition is observed in pipes of moderate length for R < lo4 due to detachment on sharp angles in the entrance section, or in the range R 104-105 due to the instability of the boundary layer close to the wall of the pipe. At any rate, the globally subcritical character of the transition toward nontrivial flow regimes, that implies coexistence in phase space, is essential to understand the coexistence of turbulent and laminar flow in physical space with well-defined fronts separating them at one and the same value of the control ~ a r a m e t e r . ’ ~ N

6.3.4.2

Turbulent spots in Blasius and Poiseuille flows

Described first by Emmons (1951), turbulent spots developing in the Blasius boundary layer flow have the shape of an arrow head (Fig. 6.22, left).26 They move rapidly in the flow, about 90% and 50% of the speed a t infinity for the head and the tail respectively. They widen while advancing with a spreading angle of order 10”. They also thicken by entraining laminar fluid from the outside of the boundary layer. In order t o partially suppress problems linked to the space dependence of the base flow, Carlson et al. (Note 19, p. 250) have studied the development of spots artificially produced by local triggering in plane Poiseuille flow ( R is now independent of the streamwise coordinate x). As already mentioned, turbulent patches persist within laminar flow for R 2 1000, i.e. much below the value at which nonlinear TS waves bifurcate (R,1 N 2900). As seen in Fig. 6.22 (right), triggered spots are oval. The head and tail of a spot move a t speeds roughly 2/3 and 1/3 of the centerline velocity, re25Y. Pomeau, “Front motion, metastability and subcritical bifurcations in hydrodynamics,” Physica D 23 (1986) 3-11. zsB. Cantwell, D. Coles, P. Dimotakis, “Structure and entrainment in the plane of symmetry of a turbulent spot,” J . Fluid Mech. 87 (1978) 641-672.

6 . Open Flows: Instability and Transition

257

Fig. 6.22 Turbulent spots in the Blasius flow at high R (left, after Cantwell et al. in [van Dyke (1982)]), and in plane Poiseuille flow (right, after Carlson et al., Note 19).

-

spectively. The spreading angle is of the order of 8”. Further downstream, at distances 150h ( h = channel’s height), a typical turbulent patch has a diameter of order 30-50h and then experiences spot splitting: a calm region appears, separating two turbulent spots that run side by side and further grow. Turbulent patches that develop spontaneously within the laminar flow at high levels of residual turbulence evolve similarly but have less regular shapes and, of course, appear randomly in space and time. As a general rule, turbulent spots display different regions that are well visible in Figure 6.22: (1)the spot as a whole, a turbulent region with welldefined boundaries, that behaves as an obstacle in the flow since it moves somewhat more slowly than the neighboring laminar flow; ( 2 ) on the sides, trains of oblique waves seemingly produced by the motion of this obstacle; (3) an interior filled with small scale turbulence; and (4)a turbulent “wake” made of streamwise elongated streaks. These features are also present in Figure 1.7, p. 17, displaying a mature turbulent spot in plane Couette flow. 6.3.4.3

Plane Couette flow

Plane Couette flow, conceivably the simplest case of wall flow, is known to stay linearly stable for all R (Note 23). This guarantees the subcritical character of the transition, avoiding any risk of confusion with scenarios based on saturated TS waves. By contrast with Blasius or Poiseuille flows, there is further no problem with the downstream transport of spots when using the apparatus used in S a ~ l a and y ~ ~described in Figure 6.23 since, by construction, the mean advection cancels exactly for symmetry reasons. In 27F.Daviaud, J . Hegseth et P. Bergd, “Subcritical transition to turbulence in plane Couette flow,” Phys. Rev. Lett. 69 (1992) 2511-2514; 0. Dauchot and F. Daviaud, “Finite amplitude perturbations and spot growth mechanism in plane Couette flow,’’ Phys. Fluids 7 (1995) 335-343.


258

tv

guiding rolls

Fig. 6.23 Setup used to produce high-quality plane Couette flow in view of a quantitative study of turbulent spots. Left: The shear is created in the gap separating two portions of a wide plastic belt kept parallel at a constant distance 2h by four adjustable rolls, with counter-plates limiting mechanical vibrations, and entrained at linear speed U by two large rolls. Right: Turbulent spots are generated by a thin jet that goes through the flow when holes in the belt and the counter-plates are aligned. After Daviaud and coll. (CEA Saclay).

fact, it is not so easy to produce a flow with a low level of residual turbulence and sophisticated image processing is necessary to reach a meaningful quantitative leveL2* Traditionally, the Reynolds number is defined as R = U h / v where U is the speed of the wall inducing the shear and 2h the width of the gap between the walls. It turns out physically more significant to define R from the shear rate U l h , the inverse of the advection time, and the viscous time over the gap, = ( 2 h ) 2 / v ,since experiments show that the structures involved in the transition occupy the full gap. - The Reynolds number will accordingly be defined as R = 4 U h / v = 4R, allowing easier comparisons with, e.g. Blasius or Poiseuille flows. Turbulent spots in plane Couette flow display characteristics similar t o those in other wall flows. It is now well established that the plane Couette flow is globally stable forz9 R < R, N 1300. This value compares well with that in the plane Poiseuille flow case (- 1000 for sustained spots), which is easily understandable when noticing that the latter can be viewed as two juxtaposed Couette flows (the centerline velocity U,, would be 2U and the half-height of the channel 2h, hence Rpois. = 4 U h / v again).

28Equally interesting information can be obtained from direct numerical simulations of Navier-Stokes equations, e.g. A. Lundbladh and A.V. Johansson, “Direct simulations of turbulent spots in plane Couette flow,” J. Fluid Mech. 229 (1991) 499-516. 29For a review of experimental results, see P. M. and 0. Dauchot, “Patterning and transition to turbulence in subcritical systems: the case of plane Couette flow,” in Coherent structures in classical systems, D. Reguera, J.M. Rubi, L.L. Bonilla, eds. (Springer Verlag, 2001), pp. 58-79.


259

Ft0’4i 1 0.2

0.0

400

300

Fig. 6.24 Turbulent fraction function of time for different values of R . ‘Q’ indicates a quench experiment, and ‘S’ a spot triggering experiment. Top: Immediate unconditional relaxation toward laminar flow. Left: Relaxation usually after a long chaotic transient. Right: Sustained turbulence. Adapted from original results, courtesy S. Bottin and 0. Dauchot (CEA Saclay, 1998).

The expression ‘globally stable’ is to be taken with precisely the meaning defined in Chapter 1, p. 9 and Figure 1.2. Experiments leading t o the result are principally of two kinds:30 either q u e n c h experiments (‘Q’)where a fully turbulent state, prepared at some large initial R, Rinit >> R,, is suddenly decreased to some final value Rfin, or spot triggering experiments (‘S’) achieved as sketched in Figure 6.23 (right). In each case, the evolution of the turbulent fraction Ft, that is the relative surface occupied by the turbulent flow, is extracted from video recordings of the flow pattern, Figure 1.7, p. 17, and appropriate thresholding of the images. As illustrated in Figure 6.24, these experiments show that for R 21 1120 all perturbations relax rapidly, whatever their initial structure, shape, and amplitude. For R between R, 21 1250 and R, E 1300, turbulent patches exist and live sufficiently long for the average measure of their surface as a function of time t o make sense. This lifetime diverges as R gets closer and closer to 30S.Bottin, F. Daviaud, P. M., 0. Dauchot, “Discontinuous transition to spatiotemporal intermittency in plane Couette flow,” Europhys. Lett. 43 (1998) 171-176.

260


sustained striped turbulence 8

I

1

I :

I

I

I

I

R

-

Fig. 6.25 Turbulent fraction Ft as a function of the Reynolds number, from a compilation of experiments.

R, from below. Finally, turbulent patches persist indefinitely with finite probability for R > R,. It can be noticed that even when turbulence is sustained (Fig. 6.24, bottom-right) some spots may relax immediately (label 3 2 ’ ) though prepared in the same way as those that do not abort (‘Sl’). This feature is the trace of both the globally subcritical character of the bifurcation, and the sensitivity to initial condition expected from a basically chaotic process. Numerical simulations confirm this point by showing that, for Reynolds numbers in the corresponding range, the border of the attraction basin of the laminar flow is fractal.31 Furthermore, in this regime, the average value of the turbulent fraction does not depend on the type of experiment (Q or S), which qualifies it as an adequate order parameter. The bifurcation diagram displayed in Figure 6.25 summarizes results on the average turbulent fraction Ft as a function of R. This diagram suggests that Ft increases regularly with R. As a matter of fact, the flow becomes uniformly turbulent beyond Rt 2~ 1660. In the range R, < R < Rt the intensity of turbulence appears to be modulated in the form of alternately laminar and turbulent oblique bands. Just beyond R,, these bands are fragmented (Fig. 6.26, top) but, as R increases, for R 1400-1450, fragments join together t o form continuous bands (Fig. 6.26, bottom). Then the modulation progressively disappears as the regime of uniform turbulence is 31A.Schmiegel and B. Eckhardt, “Fractal stability border in plane Couette flow,” N

Phys. Rev. Lett. 79 (1997) 5250-5253.

6. Open Flows: Instability and Dansition

261

Fig. 6.26 Typical oblique turbulent stripes observed in the range R, < R < Rt in the plane Couette flow for R = 1360 (top) and R = 1432 (bottom). Courtesy A. Prigent (CEA Saclay).

I globally stable

Ru G 12510

)4 2s

Be

sustained turbulent stripes g

1440

00,

1

I

featureless turbulence

R g ~ 1 3 0 0 Rt=1660

53

Fig. 6.27 Stability diagram of the plane Couette flow.

reached. These results are recapitulated in Figure 6.27. Turbulent bands are strikingly similar to the celebrated turbulent spirals observed in cylindrical geometry (Taylor-Couette flow, see Exercise 3.3.5, p. 113) when the cylinders rotate in opposite directions and sufficiently fast. Early observations of these spirals date back to Coles’ experiment^.^' A quantitative bifurcation diagram has been established by Andereck et a133 for a ratio 77 = T 0 / q = 0.883 where r , and ~i are the radii of the outer and inner cylinders respectively. Further experiments performed by Dauchot and Prigent34 with 77 = 0.983 show that bands and spirals take place in 32D. Coles, “Transition in circular Couette flow,” J. Fluid Mech. 21 (1965) 385-425. 33D. Andereck, Liu, H. Swinney, “Flow regimes in a circular Couette flow system with independently rotating cylinders,” J. Fluid Mech. 164 (1986) 155-183. 34A. Prigent, G. GrBgoire, H. ChatB, 0. Dauchot, W. van Saarloos, “Large-scale finitewavelength modulation within turbulent shear flows,” Phys. Rev. Lett. 89 (2002) 014501.

262


the same range of Reynolds numbers (weakly dependent on the average rotation rate as far as spirals are concerned), once computed, as in the case of the plane Couette flow, from the shear rate and the viscous time over the gap when curvature effects are small, Q 1. A physical explanation for the occurrence of turbulent bands or spirals is still lacking at present. The principal interest of this detailed study is t o give a concrete expression to the conceptual presentation given in Figure 6.20, especially in connection with the nature of the states on the postulated disconnected nontrivial branch of the bifurcation diagram (compare with Figure 6.25). 6.3.4.4 Mechanisms? Turbulent spots appear to play an essential role in transitional wall flows. They have common features that do not seem to rely on TS waves and their nonlinear development. First they are present at “intermediate” values of R, i.e. not too large so that the flow as a whole remains relatively coherent and not yet filled with small scale eddies as in fully developed turbulence (Chap. 7). For plane Couette flow this transitional regime extends in the range 1250 < R < 1440. Trying to identify a limited number of interacting modes and processes thus seems meaningful. According to the Squire theorem and the conventional TS transition scenario, the most dangerous infinitesimal perturbations are two-dimensional, i.e. periodic in x, functions of y, and independent of z. Disturbances relevant to the non-conventional scenario are thus expected to have finite amplitude and be mostly spanwise, i.e. depend on z (and y) but not on x. Small perturbations of this kind are damped in the long term but transient linear amplification is not excluded owing to the non-normal character of the linear stability operator (see Appendix A, 5A.3). The required finite amplitude can thus be reached provided that sufficiently large residual turbulence or intentional disturbances are present. (The linear scheme is therefore not completely abandoned since, on general grounds it is hopeless to find nontrivial nonlinear solutions from scratch.) In practice it is not too difficult to imagine a mechanism by which streamwise structures can be sustained in the flow. As a matter of fact, a streamwise vortex [0,v(y, z ) , w(y, z ) ] , even transient and weak, induces a redistribution of streamwise velocity b U ( y ,z) by displacing fluid with speed U(y) to a different cross-stream coordinate y. This perturbation bU corresponds to a slowing down or a speeding up, depending on the local direction of displacement of the basic flow lines by the vortex (see


263

Fig. 6.28 Streamwise vortices and formation of streaks.

Figure 6.28). The presence of long, well visible, streamwise streaks in the flow is thus not so surprising. The superposition of the base flow and the streamwise component of the induced flow now displays inflection points that make it unstable to KH-like mechanism producing z-periodic waves, which in turn can trigger the transition. The question is then to determine the initial amplitude of streamwise vortices able to do the job (to find the boundary of the attraction basin of the base state in dynamical systems terms). The answer has been searched mostly by direct numerical simulation of the NS equations and comparison of scenarios starting with different initial perturbations. For a detailed presentation with references, consult [Schmid and Henningson (2001), Chap. 91. This points to the receptivity problem, i.e. the way the flow converts residual turbulence or controlled disturbances into effective perturbations. In boundary layers, perturbations in the form of oblique waves (an anticipation of secondary instability modes of TS waves) seem to be most efficient. In the case of plane Couette or Poiseuille flows, with Reynolds numbers independent of the streamwise distance, the result is often presented in the form of a diagram relating the initial amplitude required for transition and the Reynolds number in the form

where y is an empirically determined exponent and A is a measure of the perturbation amplitude in the sense of Figure 6.20 (essentially the squareroot of the kinetic energy contained in the disturbance). The above relation is supposed to hold for R >> 1. This does not raise questions in the case of plane Couette flow which is linearly stable for all R. In the plane Poiseuille flow case, such a relation can be strictly valid only in the subcritical regime,

264

Instabilities, C h a o s a n d Turbulence

R < R, E 5772, since beyond it is linearly unstable, but fortunately R, is quite large and turbulent patches are expected beyond R 1000, which leave some room to the observation of the scaling behavior. For boundary layers, the relation makes sense only as long as the non-conventional transition through turbulent spots a t high levels of residual turbulence preempts the conventional TS transition. The value of the exponent y depends on the flow considered and the type of perturbation introduced. Values such that y FZ 514 (plane Couette) and FZ 714 (plane Poiseuille) have been extracted from direct numerical simulations of the NS equations.35 A functional approach developed by Chapman36 leads to y = 1 for plane Couette flow and all kinds of perturbations, and to y = 312 or y = 514 for plane Poiseuille flow and streamwise vortices or oblique waves, respectively. It can be noticed that analytical values are slightly smaller than experimental ones, which means that the considered perturbations are effectively more dangerous than estimated (their amplitudes decrease faster than predicted when R increases). The study of processes leading to the transition can also give ideas about the nature of states belonging to the nontrivial branch since it is sufficient to admit that the instability of the streamwise streaks produces new streamwise vortices, which closes the loop. Such a cycle has indeed been pointed out in numerical simulations. The model studied considered in Exercise 4.6.6 with its variables X and Y representing the amplitude of the streaks and the streamwise vortices respectively, possesses both trivial and nontrivial states, but is too simple t o account for a realistic feedback. A more sophisticated model has been proposed by Waleffe37 coupling four variables. It reads:

-

U = -rc,R-’U

+ u,MV,

- uwW2

+ uVW2, w = - K ~ R - ~ +WU,UW - U,MW V = -rcVR-lV

M = -K,R-~(M - 1) + o,w2

-

- U,VW,

U~UV,

where M represents the mean flow, V the amplitude of the streamwise vor35S.C.Reddy, P.J. Schmid, J.S. Baggett, D.S. Henningson, “On stability of streamwise streaks and transition thresholds in plane channel flows,” J. Fluid Mech. 365 (1998) 269-303. 36S.J. Chapman, “Subcritical transition in channel flows,” J. Fluid Mech. 451 (2002) 35-97. 37F. Waleffe, “On a self-sustaining process in shear flows,” Phys. Fluids 9 (1997) 883-900.

6. Open Flows: Instability and R a n s i t i o n

265

tices, U that of the streaks, and W a supplementary variable accounting for three-dimensional features of the perturbed flow. Constants K. describe the viscous effects and coefficients u are coupling constants arising from the NS advection term. Their values are such that the kinetic energy is conserved in the sense of Exercise 2.5.2. In addition to its base state ( M = 1,U = V = W = 0), the model has several nontrivial states for R sufficiently large, representing possible operating points of the cycle mentioned above. In particular, in the perturbation equations around the base state, the equation for U contains a term in V (from the linearization of the term n,MV) that accounts for the growth of the streak component induced by the streamwise vortices at the origin of transient energy growth. The boundary of the attraction basin of the base state has been studied in detail for this system and complex transients reminiscent of what is observed in laboratory experiments below the global stability threshold have been shown to exist.38 More realistic modelings are the subject of ongoing work. In the following we will no longer consider the instability mechanisms at the origin of unsteadiness in turbulent flows but rather focus on the statistical aspects of the flows that develop beyond the transitional regime. 6.4

6.4.1

Exercises

Velocity-vorticity perturbation equations

Starting from Equations (6.17-6.20) for the perturbation, show that the pressure fulfills a Poisson equation: v2p = -2va,u

(6.52)

by computing the divergence of the momentum equations and using the continuity equation. Applying the three-dimensional Laplacian to the equation for v, next eliminate the pressure variable thanks to (6.52) and show that this yields:

(at + UdX)V2V- d,,U~,V

= R-lV4v.

(6.53)

Finally, by cross differentiation and subtraction of equations for u and w show that the equation for R, = a,u - a x w reads (at

+ Uax)Rg - R-lV2Ry = -ayUa,v.

(6.54)

380. Dauchot, N. Vioujard, “Phase space analysis of a dynamical model for subcritical transition to turbulence in plane Couette flow,” Eur. Phys. J. B 14 (2000) 377-381.

266


Notice that (6.53) is formally identical t o the Orr-Sommerfeld equation (6.30) and is closed for v, while (6.54), called the Squire equation, is not closed for R, but contains a forcing term in v. Show that the the no-slip condition u = v = w = 0 at a solid wall implies the boundary conditions v = dyv = 0 and R, = 0. 6.4.2

Derivation of the Squire theorem

The theorem states that for a one-directional plane flow vo = U(y)jZ (see Figure 6. la), the most dangerous perturbations are two-dimensional, i. e. without spanwise velocity component and independent of the transverse coordinate z .

1) Write down the linearized NS equations (6.17-6.20) for a normal mode taken as [v,P](x,Y,6t)= [V,P](Y)exP[i(kx(x - c t ) + & z ) ] *

2) Notice the symmetrical role of k x u and k,w in the continuity equation and derive the equation governing kxu kzw by appropriate combinations of the equations for u and w .

+

+

3) Check that the equations for k,u kzw and v are those of a twodimensional problem in x and y (6.24-6.26) for some wavevector k and a Reynolds number R to be identified. 4) Conclude the argument by comparing the critical Reynolds numbers corresponding to a three-dimensional mode ( k x ,ky) and the associated twodimensional mode (& 0) in the viscous case, and the growth rates of these modes in the inviscid case. 6.4.3

Derivation of Fjflrtoft’s theorem

Rayleigh’s theorem is obtained by working with the imaginary part of equation (6.34). Fjortoft’s improvement is gained by manipulating its real part.

1) Use Rayleigh’s theorem to show that the real part of the phase velocity U in the real part of (6.34).

c, can be replaced by any real speed

2) Choose the speed U = Us = U(ys) a t an inflection point (ys such that U”(y,) = 0) as a reference speed and expand U(y) - Usand Uf‘ in Taylor series in the neighborhood of ys. 3) Insert these expansions in the identity for the real part of (6.34) and,

6 . Open Flows: Instability and Dartsition

267

assuming that the velocity profile U(y) is monotonic with a single inflection point, derive a necessary condition for instability by the argument used t o obtain Rayleigh's criterion. Express the result by noticing that U'(y) is the opposite of the base flow vorticity, (V x v), = axvy - ayvx, Fig. 6.4. 6.4.4

Stability of some linear-by-part velocity profiles

Instability modes of profiles in the form (6.36-6.39) are searched here by direct computation. (For unbounded domains, the boundary conditions at infinity are obtained by expressing that the solutions do not diverge exponentially.)

6.4.4.1 Derivation of matching conditions (6.38) and (6.39)

1) Express the continuity of pressure from equation (6.24) for the streamwise velocity component u at discontinuity points of U and/or U'and derive condition (6.38). 2) Write down the equation governing the motion of a material point with cross-stream coordinate Y belonging to the interface between the two flows, first in full generality and next for a normal mode (ie., for Y = I'exp(ilc(x - c t ) ) ) . Noticing that by definition of the cross-stream velocity component one has dY/dt = v, obtain as a function of and derive the matching condition (6.39).

4

6.4.4.2 Kelvin-Helmholtz instability and mixing layer 1) Find the dispersion relation of normal modes for a confined mixing layer with velocity profile:

where ymin < 0 and ymax > 0 are the cross-stream bounds of the fluid vein.

2) Same question for the smoothed velocity profile:

Take Ymin = -Ymax. Consider first ymax large and next ymax > 1 but finite.

268


3) Same question for the monotonic profile defined as

U(Y)= c"Y +

- 4(lY

+ 11

-

IY - 11)

on the interval [-b;b], b > 1 and discuss the solutions according to the value of a. a) Show that, for all a,modes with k + 00 are not unstable. b) Show that there cannot be unstable modes when a > 1. c) For a < 1, show that there exist unstable modes when k + 0 and a M 1 (the complete dispersion relation should be determined outside this neighborhood). Interpret these results in view of Fjortoft's theorem.

6.4.4.3 Stability of the plane j e t The base velocity profile of a plane jet (with speed U = 1) of some inviscid fluid entering a large vessel of the same fluid at rest can be taken in the form

< -1,

U(y) = 0

for y

U(y)=l

for

-l

3The risk of confusion of this quantity with a wavevector is small in the present context. The presence of term $ k & j in expression (7.30) guarantees that the trace of the tensor is zero, owing to the mean flow incompressibility condition, 8;K = 0.

7. Developed Turbulence

293

that has been extended to general flows by Smagorinsky (1963) in the form (7.32) The two formulations are easily checked to be equivalent when used in Prandtl's conditions: evaluating the second form with a mean strain tensor Sij reduced to S l 2 = 5'21 = i a l V 2 , one gets: vt = [ 2 ( S 1 2 S 1 2 + S 2 1 S 2 1 ) ] 1 / 2 = [4(plv2)2]1/2

=

p1v21.

All this however does not make a complete model since an expression for l m i x is still requested. It is thus advisable to turn to complete models, still resting on the eddy viscosity assumption, but free of external specification of vmix and/or Cmix on a case-by-case basis. These models can be developed at several levels depending on the number of additional dynamical equations closing the Reynolds averaged equations. From this viewpoint, mixing length models are zero-equation models (or 'algebraic' models). One-equation models determine the velocity scale Zmix from the turbulent kinetic energy k by setting Zmix 0: A. The latter quantity is then obtained from Eq. (7.24), with appropriately modeled unknown terms. This equation is rewritten here in the generic form: (7.33) where 3(", P(') and D ( k ) represent flux, production, and dissipation terms, respectively. The actual flux term is replaced by a conventional diffusion term:

where the turbulent diffusivity of energy Kt is related to the eddy viscosity (turbulent diffusivity of momentum) vt by means of an effective Prandtl number traditionally noted f f k , i.e. /st = vt/ok. In full generality, the production term reads P = -(i.jii.jj)Sij. The eddy viscosity assumption turns it into:

which is positive definite. The dissipation term, corresponding t o the energy E transfered from the large scales is modeled by (7.1), which yields:

Instabiiities, Chaos and m r b d e n c e

294

where CD is one of the constants in the model. The presence of Cmix in that equation, as well as in the relation that defines the eddy viscosity, that is vt 0: k1/2.tmix,still makes this type of modeling incomplete. This limitation is raised by adding a supplementary equation. The best known two-equation model is the k-E model that collects all that precedes but, instead of adding an equation governing Cmix, uses the fact that the missing length can be constructed from k and E as Cmix 0: k 3 l 2 / c , which is just Eq. (7.1) solved for C given E . The eddy viscosity is then taken as: k2 ut =

c,-, €

where C, is a constant. The equation for E that closes the system is copied from (7.33): (at

+ ?a,)€ = a.F(.) + P(‘) - d E, ) 3 3

(7.34)

whose different terms are modeled in a fully parallel way:

g“)= -$)VE, with K?) = u t / u f ,where u, is the Prandtl number relative t o the “diffusion” of E . The production and dissipation terms are taken further in the form

which can be justified by noticing that quantity k/E is homogeneous to a time T and that a dimensionally consistent way t o pass from (7.33) t o (7.34) is by dividing (7.33) by this time and introducing phenomenological dimensionless constant to account for the differences between the processes governing the dynamics of k and E . Constants appearing in the model, c,, cE1,c , 2 , f f k and u, are generally adjusted by demanding that the predictions in few test cases (decaying turbulence, log law in boundary layers, etc.) correspond to empirical data. To go further one should consult [Pope (2000)], Chapter 10. 7.4.2

Large eddy simulations

The variables introduced above refer to ensemble averages. However, quantitative information collected from the study of specific realizations (structure and temporal history) are at least as important as the properties of statistical ensembles ideally corresponding to these experiments. There is

7. Developed TuTbu[ence

295

therefore a pressing need for numerical simulations of Navier-Stokes equations at large Reynolds numbers, in industrial or geophysical configurations for example. The evolution of small scales is of little interest, except that they have t o be present to guarantee energy dissipation. On the contrary, large scale motions that carry the energy and achieve the mixing are of more interest. This suggests performing a filtering of the equations that eliminates small scales, as implemented in large eddy simulations (‘LES’). The need for a filtering of primitive equations is clearly understood from Landau’s estimate (7.8) of the number of degrees of freedom relevant t o developed turbulence, which might open a pessimistic perspective: despite the present observed increase of computational power, Reynolds numbers accessible to direct numerical simulation of Navier-Stokes equations will, for long, stay modest (see Exercise 7.5.1). Considering spectral approaches to the simulation problem (cf. Appendix B, 5B.2.2) will allow us t o better understand why and how one can develop an alternative strategy. In practice large scale eddies are reconstructed from a set of small wavevectors of the order of the inverse of l o , the scale in which energy is injected in the flow by the main instability mechanism. Few steps in the cascade are necessary t o reach complete disorder. These steps depend little on the precise value of the Reynolds number. Let us call4 ~ L Ethe maximum wavevector necessary to account for large eddies (hence subscript ‘LE’). One thus gets ~ L 0 E: ko, with a rather small proportionality factor, typically 2n for n steps, with n = 3 or 4. In order of magnitude, the number of such modes is thus given as the volume of a sphere with radius ~ L in E , wavevector Fourier space: NLE c( kiE. By contrast the value of k ~ the associated to the Kolmogorov scale (7.6) marking the end of the inertial cascade, is strongly dependent on R : kK 0: R3I4. The corresponding number of active modes, given by the Landau estimate Ndiss 0: ki R9I4, is rapidly diverging. Let us denote k,, the largest wavevector included in the simulation (typically, k,, = ~ r / A x ,where Ax measures the space resolution). In principle the injected energy will be correctly dissipated a t the given value of R if kmax 2 k~ and thus N,,, 0: k,,3 2 Ndiss, which is impossible for the values of R of interest in applications. In fact, the large scales will be correctly described provided that the energy is correctly transferred through the scale derived from ~ L E . Accordingly, simulating the whole range of scales between k L E and k~ in the inertial cascade is necessary only if one does not know how t o parame-

-

4From now on k no longer refers to the fluctuating kinetic energy but serves again t o label wavevectors.

296


-

terize this transfer through lc l c ~ ~The . filtering is thus an operation by which numerous “useless” modes (Ndiss- NLE >> N L E )are eliminated t o the benefit of few modes below ICLE “useful” t o the description of large scale motions. A detailed operational presentation of the method is out of question within the limited scope of these notes -for this one may consult [Pope (2000)], Chapter 13- but just give a qualitative idea of it, resting on, and extending the discussions in the previous subsection. The ensemble averaging leads to a splitting of the hydrodynamic fields into two components (7.14) that fulfill (7.15). Here the flow is similarly decomposed into filtered quantities (v,p)f (superscript ‘f’) and residual quantities (v, p)‘ (superscript ‘r’). The filtered velocity field in physical space is obtained from the convolution of the primitive field with a filter function GA:

where GA(Y,X) is some appropriately normalized integral kernel GA(Y,x) d3y = 1) that is localized in space over a region of typical size A. The filter is homogeneous if it does not depend on position x which is usually the case except close to a wall). Furthermore it is isotropic if it depends only on the length IyI and not the orientation of y. The simplest homogeneous isotropic filter corresponds t o the averaging over a sphere with diameter A (the averaging over a cube of side A would not be isotropic since the result would depend on the local flow direction with respect to the coordinate axes). Smoother filters, e.g. Gaussian, are usually considered. Inspired by the initial remark about modes in Fourier space, one can also work in spectral space and perform the filtering in that space where the convolution product of two functions in physical space is just the ordinary product of their Fourier transforms. The Fourier transform of the filter function in physical space is called the transfer function. The transfer function of a Gaussian function is another Gaussian function, whereas the transfer function of a square window displays damped oscillatory wings in Fourier space. Conversely, the filter function of the low-pass square filter in spectral space also displays wings in physical space. The simplest spectral filtering is indeed achieved by such low-lass filtering that amounts to projecting the dynamics onto the subspace of Fourier modes such that lc 5 l c ~ where , l c is ~ the cut-off wavevector (i.e. truncating the Fourier decomposition of the fields beyond k ~ ) .

(s


297

The residual velocity field is by definition the difference between the primitive field and the filtered field:

t ) = v(x, t ) - Vf(X,t ) .

V'(X,

The filtering is somewhat less simple than the ensemble averaging. First it is to be noted that except when it comes to a projection over a set of modes, and contrary to averaging (see (7.15), p. 281), filtering is not an idempotent operation. Applying the filter to an already filtered quantity does not leave it unchanged, since filtering a residual does not give zero. One gets: v

f

G

v +v'

+

vf = [v']

f

+ [v']

f

with

[v']

f

$ 0 (in general).

In the same way, it is easy to show that time differentiation commutes with filtering but that space differentiation commutes only if the filter is homogeneous, which is shown from a direct computation starting with (7.35):

Except in special cases (e.9. close to a wall) one can work with a homogeneous filter and try to get, in this simple case, the equations governing the filtered fields to be explicitly resolved in a simulation. Filtering the continuity equation immediately gives:

[a&

f

=

a,.,'

=0,

and, by subtraction:

for the residue. Difficulties with the momentum equations are analogous to those that led to the introduction of the Reynolds stresses, p. 282. In full generality, these equations read: p (atwf

+ w,'a,wf)

=

a,

(Ufi

+ Ti'j) ,

(7.36)

where ut is the stress tensor defined in the usual way with the filtered quantities:

uf. 23 = -pf&

+qsf. 23

with

sf, =

(aiw,' + ajvzf) .

The supplementary term T : ~ plays the role of the Reynolds stress in the ensemble averaging approach. Its presence is due to the fact that the term


298

[vivjlf that appears upon filtering the advection term is not equal to the product v:v; of the filtered components of the velocity. Introducing the decomposition v = vf v’, one readily gets:

+

f [WjVi]

=

I:.;[

I;.+;.[ f

f

+ [vpzf]f + [v;.;]‘

.

(7.37)

The first term on the r.h.s. cannot be expressed as a product of resolved f (i.e. filtered) quantities but the difference [v;wf] -v;vf, called the Leonard stress, is usually quantitatively small. The two terms next involve resolved and residual scales, and the last term, called subgridscale though at this stage no computational grid has been introduced, involves non-resolved scales only. A closure problem of a nature somewhat different from that in ensemble averaging is then posed. It rests on a modeling of the additional tensor T,& in (7.36), which is further decomposed into an isotropic part that contributes t o modify the filtered pressure and an anisotropic residue. This anisotropic part is most often modeled using an eddy viscosity assumption that closes the system at this level. A Smagorinsky formula (7.32) may be chosen, where the strain tensor is of course defined in terms of the filtered fields. In spite of an apparent parallelism, this approach is rather different from that developed in the previous subsection. As a matter of fact, here the filtered fields are fluctuating quantities while before they were averaged non-fluctuating quantities. The presence of time in equations (7.21-7.24), and later in the lc-E model (7.31-7.34) has indeed a different meaning related to the specific nature of the ensemble statistics. More or less sophisticated models -here we have just suggested the simplest formulation- used in LES present themselves as systems of equations for a “turbulent fluid” with specific rheological properties that would depend on the local state of the flow. They are still largely under development and demand to be validated by comparisons with direct simulations (moderate R) and laboratory experiments (high R ) , since they settle the actual complexity of turbulent exchanges, cf. (7.37), via a closure assumption. The strategies used to solve the problem of interaction between resolved and residual motions might inspire the scientific approach t o the nonlinear dynamics of complex systems such as the climate system to be considered as an example at the end of the next chapter, after a brief summary of the topics touched on up to now.


7.5

7.5.1

299

Exercises

Scales in turbulence and numerical simulations

Direct numerical simulations of Navier-Stokes equations in a situation of developed turbulence are considered, in view of the amount of computer resources required for reliable results as a function of the nominal Reynolds number. The linear size of a cubic domain where a turbulent flow is L , the typical velocity U , v is the kinematic viscosity and we assume that the Reynolds number can be varied through Y at given L , U . 1) Recall Kolmogorov’s estimate of the dissipation scale CK relative t o L at given R. 2) From a numerical point of view, but without reference to a specific numerical scheme (as discussed in Appendix B), Ax and At denoting the space and time integration step, respectively, it is assumed that the small scales are correctly resolved provided that CK 2Ax. Determine the variation with R of the number of grid points needed to resolve the smallest scales. 3) Numerical stability considerations come and limit the time step At. In a system dominated by advection (term v . V())the stability is controlled by a CFL (Courant-Friedrich-Lewy) stability criterion stipulating that the propagation speed of local (numerical) information A x / A t must remain larger than the speed of transport of physical information by the velocity field. Numerical stability thus requires At < Ax/U. Derive the variation of the number of time steps necessary to perform a simulation over a time interval T with space step Ax, with T L / U (turn-over time).

-

N

4 ) Evaluate the amount of computational power as a function of R in a typical full 3D simulation, and the relative increase of power implied by a doubling of R. 7.5.2

E f i c i e n c y of turbulent mixing.

1) A radiator is installed to heat up a room with typical size L (volume L 3 ) . Determine the time necessary to reach thermal equilibrium by assuming pure thermal diffusion using a dimensional argument. Take L = 5 m, and Y = 15 m2/s. For a gas, the Prandtl number is Y / K 21 0.7. 2 ) Consider now turbulent heating with largest eddies of the size of the room

generating smaller eddies driving the process. Still dimensionally, estimate the order of magnitude of the air speed above the radiator by assuming


300

the kinetic energy is converted from the potential gravitational energy (buoyancy) over a height h above the radiator. Take g = 10 m/s2, a temperature increase 68 = 10" above the ambient temperature, h = 0.1 m. The thermal expansion coefficient is a = 1/300 K-l. 3) A fraction of this energy is available to stir the air in the room. Assuming that the effective stirring speed is only 20% of the speed computed above, determine the characteristic stirring time for the scale of the room.

7.5.3

Developed turbulence in a cumulus cloud

--

1) Consider the motions inside a typical small cumulus with diameter of the order of l o 300 m. Assuming that the speed of eddies at this scale is Vo 3 m/s, compute the corresponding Reynolds number ( u = 15 m2/s). 2) Estimate of the energy dissipation rate per unit mass and the total power (in kW) dissipated in the cloud ( p = 1.25 kg/m3). Evaluate the internal scale CK at which energy is effectively dissipated by viscous friction. 3) An observation plane goes through the cloud at speed 180 km/h, carrying a hot-wire velocity probe. This apparatus is made of a small length sensor of length 61 = 0.5 mm. Assuming that it can resolve velocity fluctuations at scales of the order 361, what is the maximum frequency of turbulent signals that can recorded by the device. Like in grid turbulence, one can suppose that the relative velocity of the probe with respect of the fluid is the essential factor in converting spatial fluctuations into temporal fluctuations according to Taylor's frozen turbulence hypothesis. Compare the Kolmogorov scale CK to the scale of the correctly sampled fluctuations. Estimate the turn-over time of large eddies and justify the hypothesis.

7.5.4

Decaying turbulence

Consider a container with volume L3 containing an initially well stirred fluid that is left to settle. With C = 1 m, and u = 151OP6 m2/s, and assuming initial velocity fluctuations of order GO = 1 m/s (V denotes here the root mean square velocity in one direction of space), compute the Reynolds number a t the beginning of the experiment. 1) Assume that the characteristic scale of turbulent motions remains C and derive the evolution equation for the kinetic energy $V2, the evolution law of the velocity and the decay law of the Reynolds number that follows.

7. Developed ‘ h r b d e n c e

301

2) Compute the time necessary to reach R 10 and derive the corresponding fluctuation velocity. At this stage, the viscous dissipation of the large scales is no longer negligible and one must abandon the inertial expression of E to, replace it with the viscous form cviSc. = cuV2/C2. Determine the value of the constant C that assures the continuity of the matching of the two laws at R = 10 and the law governing the final stage of settling, N

7.5.5

Turbulence in the atmospheric boundary layer

The atmospheric boundary layer is submitted t o the Earth’s rotation. The angular velocity of a frame linked to the surface is given by R sin(A) , where X is the latitude. (Coriolis acceleration is given by 2fl x v,, where v, is the relative velocity; f = 2R sin(X) is called the ‘Coriolis parameter’.) 1) Compute the time scale corresponding to this motion at latitude 45’ and next, by a dimensional argument, the thickness of the atmospheric layer that feels the friction on the ground in the laminar case. Take u = 15 x lop6 m2/s. 2) In fact the atmosphere is turbulent. Estimate the thickness Cmix of the boundary layer affected by turbulent fluctuations of the order of 3% of wind velocity (typically 10 m/s) over the same time scale. Determine the corresponding eddy viscosity given by ut = Gmix Cmix.

7.5.6

Two-dimensional turbulence

The spectrum of velocity fluctuations in the two-dimensional case is different from that in three-dimensional turbulence. In the latter case, the energy cascade toward small scales generates the Kolmogorov k - 5 / 3 spectrum. In two dimensions it is admitted that the turbulent processes are controlled by the transport and dissipation of vorticity. In two dimensions the vorticity has a single component w = axuy - ayux (along the normal t o the plane in which the motions are supposed to take place). The enstrophy is this then defined as W = ;w2. 1) Consider an eddy of arbitrary size C and velocity Ue, estimate its vorticity we and the associated enstrophy. 2) Let W I be the vorticity present in the injection scales CI and assume that, per unit time, eddies loose a fixed fraction of the enstrophy present at these scales. Paralleling the argument for energy injection in three dimensions, estimate the rate of injection of enstrophy per unit time.

302


3) During the cascade, the size of an eddy is reduced but its vorticity is conserved. Evaluate the typical velocity ve function of WI and 1. 4) Dimensionally, the energy spectrum E ( k ) is an energy per unit wavevector (and per unit mass). Show that the inertial enstrophy cascade leads to the Kraichnan spectrum E ( k ) 0; p 2 / 3 k - 3 .

Chapter 8

Summary and Perspectives

Through all these pages, our main aim has been to present an introduction to the study of complex systems from the viewpoint of nonlinear dynamics. An abstract approach generalizing the classical treatment of oscillators has been developed and applied to several concrete situations encountered in hydrodynamics. The validity of this theory eventually rests first on the recognition of space-time macroscopic coherence induced by instability mechanisms that optimally develop with specific scales, as opposed to the incoherence of motions involved in plain relaxation to thermodynamic equilibrium at microscopic scales. In turn this coherence reduces the multiplicity of possible dynamical behaviors, up to a point where the context of low-dimensional dynamical systems become an appropriate setting. Complexity then mainly means chaos, temporal or spatio-temporal. Such instabilities are generic in macroscopic systems driven far from thermodynamic equilibrium by external stresses. The tendency to return to uniform local configurations above implies a resistance to change expressing the dissipative character of the global dynamics, which is not incompatible with slow large-scale unsteady evolution. In fluid mechanics, this has been illustrated by the emergence of behaviors, from well-ordered cells or waves, to randomly disorganized, turbulent eddies. After a brief summary of the results obtained so far (58.1, 8.2) we examine whether the approach that has been followed can be transposed to other situations where “aggregates” of different nature are in interaction over some range of spatio-temporal scales. Leaving aside applications of technical interest (e.g. flows around obstacles with complex geometries), we focus the discussion on a topic where problems already arise at the modeling stage and next when one wants to draw reliable conclusions from this modeling. Macroeconomics evolution or biodiversity dynamics in ecology

303


304

belong to this class of problems. society is the problem of climate $8.3, mostly since it is still rather it might not be completely free of 8.1

Of equal interest to the future of our change, which we choose to consider in close to physics and mechanics (though geopolitical afterthoughts).

Dynamics, Stability, and Chaos

The concept of dynamics governing a time evolution has been modeled as an initial value problem for a differential system written in the form:

$x =F ( X ) ,

X ( t ( 0 ) )= X(O) .

(8.1)

Next, some operating point called the base state and denoted here xb is supposed to be given. The notion of stability, that occupies a central position, is easier to approach in a linear context where the departure from the base state X ' = X - xb is infinitesimal, which allows one to truncate the perturbation expansions beyond the first significant order, leading to problems in the form

-&XI = L X ' ,

(8.2)

where L is a linear operator to which one can apply the familiar tools of linear algebra, and especially normal mode analysis. Normal modes are taken in the form:

X' = X exp(st), where s is the growth rate. This leads to an eigenvalue problem:

(L- S Z ) X = 0

(8.3)

yielding a series of eigenvalues E C called its spectrum: s,

= un - iw,

,

n = 1 , 2 , . ..

further ordered by decreasing values of the d s : (TI 2 u2 2 ug 2 . . . . The principal result of this analysis is a representation of the dynamics in a infinitesimal neighborhood of the base state in terms of a superposition of normal modes Xn, each with an amplitude A , ( t ) , function of time and A,(t(O)) = A t ) . In the simplest cases, i.e. non-degenerate, this superposition can be written as:

X'(t)=

n

Aio)exp [ s ( t - t ( ' ) ) ] X n .

8. Summary and Perspectives

305

The stability of the base state Xb can thus be discussed as a function of the sign of the real parts (T, of the complex growth-rates s, of the eigenmodes. From the linear viewpoint a given mode n is thus stable when u, < 0, neutral or marginal when u,,= 0, and unstable when u, > 0. A suficient condition for the instability of Xb is thus (TI > 0. The imaginary part w, of s, next allows one to distinguish between stationary modes with w, = 0 from oscillatory modes with w, # 0. In full generality, F ,x b , C and its spectrum, are functions of control parameters accounting for the different ways of acting on the system, which we symbolize by a single variable T in the present formal setting. A bifurcation takes place when the real part of the growth rate of the most dangerous mode crosses zero from below as T is varied. The critical condition is thus given by ( T ~ ( T )= 0 which can be solved for T t o define the instability threshold rC. When the base state is unstable, the linear analysis is valid only as long as the system stays sufficiently close to its base state, which does not hold long since some perturbations diverge exponentially. Terms neglected until now in the perturbation expansion must be taken into account, expressing the interaction of the unstable modes between themselves and with all other modes. In this domain, the essential result is the possibility t o eliminate the stable modes Xs (‘s’ for ‘stable’), with amplitudes collectively called A,, and only keep the master modes X c with amplitudes A, (‘c’ for ‘critical’ of ‘central’, i.e. either stable or unstable but near-marginal, see below or Chapter 4, 54.1.2). The reason is that the former are not independent of the latter but rather enslaved to them. This elimination, sketched in Figure 4.3, p. 120, leads one to define an effective dynamics only coupling the central modes between themselves:

Apart from during a brief transient (exponential relaxation of stable modes), the system then evolves on a manifold defined by some relation

that can usually be determined asymptotically in the long time limit and for small departure from the critical conditions by a well-defined technique called the center manifold reduction. Technically, a gap in the spectrum has to be assumed, such that us < 0


306

and minlcsl >> Icclwith a, 0. But, in practice, the domain of interest of this reduction is larger than it seems because the normal f o r m that is obtained has most of its structure imposed by the symmetries of the problem and the nature of the critical modes. This universality opens the door to a phenomenological modeling that relies only on these ingredients. This is particularly interesting because the number of possible forms and the manner to perturb them away from criticality are both generically small. Let us mention the most frequent ones. The first one relates to the bifurcation of a single real amplitude A displaying a symmetry with respect t o the change A t-+ - A . At criticality it is governed by: N

and, upon adding the most general perturbation, by:

L d t A = r A - A3 + h ,

(8.4)

where r measures the distance to the bifurcation point in the control parameter space and h is a small perturbation that breaks the initial symmetry. (In this equation r and h are two real parameters.) This bifurcation, illustrated in Figure 4.4, p. 127, is the dynamical translation of one of Thorn’s seven elementary catastrophes and was introduced earlier in the theory of thermodynamic phase transitions by Landau. The second example is the Hopf bifurcation of a pair of complex conjugate modes that accounts for the birth of self-sustained oscillations, e.g. in the van der Pol system. Close to the threshold, the normal form reads $2 = ( r - i W , ) Z - 912122,

+

(8.5)

where 2 = A, iAi is the complex amplitude of one of the two interacting modes, wc is the angular frequency at threshold, r the distance to threshold and g a complex coefficient. Its real part g, describes the saturation of the amplitude beyond threshold while its imaginary part gi, the nonlinear dispersion coefficient, accounts for the dependence of the frequency on the amplitude, a typical feature of nonlinear oscillators, see Figure 4.5, p. 129. The story then continues with the emergence of more complex behavior, still crucially involving stability considerations. The state resulting from this first bifurcation can indeed be considered as a new base state, that may become unstable with respect to secondary modes, and then tertiary modes, etc.


307

This point of view refers to a situation where an ever larger number of modes initially in the stable group (a, < 0) become critical and next unstable ((T, > 0), therefore increasing the dimension of the effective dynamics. Stationary modes (fixed point in phase space) described using a single real amplitude, e.g. (8.4),are then usually followed by periodic behaviors (limit cycle attractors) governed by (8.5), which in turn may become unstable with respect t o other oscillatory modes yielding quasi-periodic behavior (torus attractor), etc. Ruelle and Takens have shown that, contrary t o the original intuition of Landau who argued in terms of superpositions of oscillatory behaviors, complexity arises from the interaction of modes, with chaos as a nontrivial cansequence. These two competing scenarios were sketched in Figure 3.5, p. 93. In order to understand the emergence of chaos from a regular behavior, it is most appropriate t o take the Poincare' section of trajectories, a procedure that generalizes the stroboscopic analysis of forced periodic behaviors and, at any rate, reduced the initial continuous time system to an iteration (discrete time system). The essential property of chaotic systems is the instability of trajectories on the corresponding attractor, then said to be strange, which expresses the loss of long term predictability resulting from the sensitivity to initial conditions and small perturbations. This property is understood most easily when considering the dyadic iteration, the simplest model of chaotic system:

Xk+l = 2x1,

(mod I ) ,

that perfectly illustrates the (exponential/geometrical) growth of the distance between neighboring trajectories: &XI,= 2'6x0 = exp(n In 2)bXo. In the general case, this divergence can be measured by means of Lyapunov exponents that extend the notion of growth rate initially introduced to deal with the stability of regular trajectories. For one-dimensional iterations in the form

one then defines:

where { x k , k = 0 , 1 , . . . } is a reference trajectory, so that the distance between neighboring trajectories varies as (exp for k large enough. In

308

Instabilities, Chaos and Tzlrbulence

addition, strange attractors generally display a fractal structure contrasting the smooth aspect of regular attractors. Finally, it has been seen that the transition t o chaos develops through one out of a small number of universal scenarios, the most celebrated being the subharmonic cascade that leads to an aperiodic behavior at the end of a series of period doublings (period 2, 4, . . . , 2”,. . . ) with successive thresholds geometrically converging towards a finite value, as illustrated in Figure 4.13, p. 145.

8.2

Continuous Media, Instabilities, and Turbulence

For a discrete system, (8.1) is a finite-dimensional differential system. For a continuous medium, it is a system of partial differential equations. The linearized operator defined by (8.2) also contains partial derivatives with respect t o (physical) space coordinates. In order to simplify the analysis, one often considers the limit of translationally invariant systems that allows one to solve (8.3) by means of a spatial Fourier transform, by making the change

Branches of normal modes are then parameterized by the wavevector k . The relation s = s n ( k ; r ) = r n ( k ; r ) iun(k;r) is called the dispersion relation relative to mode n. The condition C T , ( ~ ; r ) = 0, once solved for r , gives the corresponding marginal stability condition:

+

r = rim,”’ (k).

As before, one is interested in the most dangerous mode, the mode with the largest CT. Let us assume that mode ‘1’ is the most dangerous for all values of k (no branch exchange) and further drop the subscript. Usually, the wavevector of the perturbation cannot be controlled and the growth rate displays a quadratic maximum around some specific Ic, called the critical wavevector. Generically the real part can thus be written as: ToCT

= T - T c - ( i ( k - kc)2.

(8.6)

In this equation, TO is the natural evolution time of the fluctuations and 1 corresponds t o a situation where the bifurcated solution remains coherent at a local scale only, typically few X,s. s Another strategy has to be employed to account for them, in terms of envelopes describing the slow, long-wavelength modulations brought to an ideally uniform nonlinear solution. Technically these envelopes are obtained through a rather boring multiple scale strategy which can be short-circuited through a phenomenological approach making an immoderate use of the universal properties of the system close to its instability threshold. This universality is linked t o its physical symmetries and the nature of the primary bifurcation. The system is then modeled by unfolding the normal form that governs the amplitude of the critical mode, thus becoming an envelope. This is achieved by adding a dynamics, often simply diffusive, driving spatial modulations to the amplitude, which yields equations of complex-Ginzburg-Landau type, see $5.4. While being restricted to the weakly nonlinear range, these theoretical developments, sketched in Chapter 5, satisfactorily account for topological defects and universal secondary instabilities. Defects (singular points in the envelopes) and phase instabilities linked to the local translation and rotation invariances in space, both contribute to the large-time long-distance disorganization of the patterns at the heart of spatio-temporal chaos as a route to turbulence in extended systems. This route to turbulence is relatively progressive and can thus be called ‘globally supercritical’ in the sense that the bifurcating state (which develops beyond the bifurcation point) stays close to the bifurcated state (which is unstable) and replaces it. Some physical systems, and especially convection even a t large aspect ratios, become turbulent in that way. The case of shear flows is relatively more delicate. As a matter of fact, one has to distinguish flows that are mechanically unstable, i.e. unstable a t the inviscid limit, from those that remain stable in this limit. The prototype of the first class is the shear layer experiencing a Kelvin-Helmholtz instability related to the presence of an inflection point in the base profile. The second class of flow is formed with those that can possibly be unstable but through a subtle process in which viscous dissipation is crucially involved, producing Tollmien-Schlichting waves. The consequences of this classification in the strongly nonlinear regime are important. In particular, whereas the flows belonging to the first class follow a globally supercritical route to turbulence that develops at moderate Reynolds numbers, in the second case, the linear instability threshold, if it exists at all, is high, so that there is room for other typically nonlinear


311

solutions to the Navier-Stokes equations at intermediate Reynolds numbers. This explains that the transition to turbulence, globally subcritical according t o this scenario also called the bypass transition, appears t o be “wilder” than in the first case. Actually, turbulent spots concentrate, locally in space amid laminar flow, that special kind of solution that does not derive continuously from the base state, see Figure 6.20, p. 254. Practical situations are often complicated: convective/absolute instability, super/sub-critical bifurcation, regular/chaotic motion, modulations and spatio-temporal chaos. A good understanding of the processes at stake can help in mastering them, that is to say, controlling the transition to turbulence. A toolbox has been presented, and its use demonstrated in some typical cases. It contains most of the implements necessary to explore the richness that surges as soon as nonlinearities cannot be neglected, whatever the field of knowledge considered. We have said little about the regime of turbulence developed that sets in well beyond the transitional region. In this circumstance, the deterministic approach is no longer of help since many degrees of freedom interact in a cascade over a large range of space scales, from the large ones where the turbulent energy is extracted from the mean flow by the instability mechanisms to the small ones where it is dissipated by viscosity effects, see Figure 7.3, p. 276. Elementary dimensional arguments allowed us to estimate the rate of transfer of the energy (per unit mass) through the inertial range where viscous dissipation can be neglected and energy distribution is ruled by Kolmogorov’s energy spectrum E ( k ) k - 5 / 3 , down t o the dissipation scale where viscous effects cease to be negligible. The irregular (chaotic) character of turbulent flows lead us to introduce statistical methods. Two fundamental components of the flow, the average flow and the fluctuations were identified following Reynolds, but the closure problem arose immediately through the definition of Reynolds stresses. A very preliminary solution to this problem was presented, in terms of an eddy viscosity, itself evaluated within a mixing length hypothesis. The approach was then used to determine the mean profile of a fully developed turbulent boundary layer, yielding KBrmBn’s logarithmic law, see Figure 7.6, p. 289. The two main results obtained by dimensional arguments, Kolmogorov’s spectrum and KBrmin’s logarithmic law are remarkably verified by experiments but remains zero-order approximations that do not thoroughly treat the problem of exchanges between eddies with different scales. Empirical divergences appear as soon as one tries to go beyond this elementary level. N

312


In concrete configurations of interest t o applications, a numerical approach is often necessary, with a crucial modeling step to deal with the closure problem (k-e model, Large Eddy Simulations,. . . ). In the context considered up to now, the origin of the complexity was mostly in the nonlinear character of the dynamics of systems with “simple” definitions. More “complex” interactions of processes can also take place, e.g. involving rotation and magnetic fields in conducting fluids (dynamo effect), combustion, etc. possibly demanding separate modeling efforts. In the concluding section we rather introduce a case where complexity arises from the interaction of a large variety of sub-systems, exchanging many physical quantities over a large range of space and time scales, the Earth’s climate system.

8.3

Approach to a Complex System: the Earth’s Climate

In the traditional geographical sense, the study of climate was limited to a description of meteorological phenomena such as temperature, wind, amount of rain, etc., that characterize the mean state of the atmosphere throughout a typical year. This meaning of course remains but, in the perspective of a rapid change induced by human activities, one has to enlarge the viewpoint to also include how this average situation arises,’ and to understand the individual dynamics of many “actors,” the atmosphere, the ocean, the cryosphere (ice caps), the biosphere (mainly the vegetation), and at, the end of the chain, our society. The main physical media involved are fluids in motion, transporting energy in different forms. Energy exchanges play an important role but are constrained by rigorous physical conservation laws. A large variety of nonlinear feedbacks and space-time scales are involved in climate phenomena, which explains why one could naively try to tackle them using our toolbox. We therefore first extend the framework of previous approaches t o evoke a few tangible applications of the concepts introduced earlier, before going to more speculative grounds. Climate can first be understood through its meteorological dimension, variable and to a certain extent random. In French, there is a sailor saying that: Mentira bien souvent/ Qui prCdira le temps,/ Mais beaucoup ’A nice introduction is [Philander (ZOOO)]; at a more technical level one can cite [Peixoto and Oort (1992)], and [Henderson-Sellers and McGuffie (1987)] for modeling issues.

8 . Summary and Perspectives

313

moins pourtant/ S’il est bon observant, which essentially means that you are certainly a lier when you predict weather but much less of one if you make good observations. Current wisdom of old sailors as expressed in this saying is thus an excellent anticipation of the concept of deterministic chaos, originating from long term unpredictability (the lie) in spite of predictable short term evolution (reasonable extrapolation of accurate observations) as it was formally introduced by Ruelle in the Seventies, not long after the informal quotation by Lorenz -a great meteorologist, by the way- in the Sixties. The key-concept of chaos theory is the indefinite amplification of tiny differences between initial conditions, that forbids long term prediction. In the case of weather, this prediction rests on the direct integration of equations governing the motions of the atmosphere. These equations are implemented in an essentially satisfactory way in the best routines of modern Meteorological Offices but at the regional scale that we consider presently, the reliability of the results depends on an accurate account of initial conditions in the bulk of the atmosphere (pressure, winds, temperature, humidity) and boundary conditions at the surface (sea surface temperature, relief,...). This data is known with a limited precision at the nodes of a sparse network of observation point, whereas the problem is set in terms of partial differential equations, which leaves aside all local events of small amplitude. Figure 8.1 illustrates the fact that radically different predictions are obtained beyond some predictability range, today

Fig. 8.1 States at day+7 (bottom line) of three simulations starting from initial conditions all compatible with the same ensemble of physical observations (top). After Palmer, in [Hall (1992)l.

314


Fig. 8.2 METEOSATimage of a storm centered at the entrance of the English Channel, September 6, 1995; @ MCtBo-fiance, uwu.meteo.fr, with permission.

still much less than fifteen days (Lorenz’s “butterfly effect”). As a first step, one indeed has to proceed to data assimilation, i.e. put data in an appropriate form to start the simulation, after some pre-treatment optimizing the short-time evolution and making the dynamical extrapolation reliable over a longer period of time, in accordance with chaos theory that tells us that the prediction (in abstract phase space) is more accurate if initial conditions are better known, but that small initial uncertainties always reveal themselves. Recent progress may lead us to think that the detailed midterm (few days) prediction of extreme events such as the one illustrated in Figure 8.2, their conditions of occurrence and their development, is within reach and can only improve in parallel with computing power which allows one to follow individual trajectories ever better and make statistics over ensembles of trajectories, while long-term unpredictability is inescapable. Keeping the dynamical systems perspective, let us mention the use of nonlinear empirical reconstruction techniques introduced at the end of Chapter 4, 54.4. Takens’s delayed coordinate method was in particular applied to the quantitative analysis of time series measuring the intensity of the El Niiio phenomenon, as an attempt to predict its evolution one or two years in advance. El Niiio is a sequence of climatic “anomalies” that affect firstly the inter-tropical Pacific Ocean.2 It is characterized by a cyclic modification of heat exchanges and zonal motions of water masses together with a swing of the pressure field between the East-Pacific and the WestPacific called the Southern Oscillation (hence the acronym ENS0 meaning 2For an introduction, consult Chapter 9 of [Philander (2000)].

315


4

7950

1955

1960

1965

1970

1975

1980

1985

I

I

1990

1995

2000

2005

Fig. 8.3 Variations of the ME1 measuring the intensity of the ENSO phenomenon.

El Niiio-Southern Oscillation). The cycle is locked on the yearly seasonal periodicity but has a variable duration between two and seven years, with warm episodes (El Niiio) and cold ones (La Niiia). The warm phase is characterized by warm waters off the coasts of Peru before Christmas, due t o a weakening of trade winds that should push these waters further to the West. In the old days the El Niiio phenomenon was considered as a gift (at the regional scale) since abundant rains made Peruvian deserts somehow fertile. Nowadays, we have a better understanding of at-a-distance effects, and intense ENSO episodes (e.g. 1982-83, 1997-98) are considered as calamities since large regions in the World suffer from them: augmented rains and floods in western South America, drought and forest fires in the West-Pacific and Australia, not mentioning suspected consequences on the global climate. The intensity of the phenomenon is measured by several indices3 (Fig. 8.3). The aim of the game is, starting from the time series of a given index, to predict the occurrence of an intense episode six months or one year in advance. Adaptative filtering techniques (singular spectrum analysis, Note 22, p. 164) and nonlinear techniques of dynam3The first one, and the oldest is the Southern Oscillation Index, SOI, that directly measures the pressure difference between Darwin, in the north of Australia and Papeete in Tahiti (see e.g. the web site of the Australian Commonwealth Bureau of Meteorology, that displays monthly values of this index from 1876 on). Another one, produced by the Japan Meteorological Agency (JMA, cftp: //uww coaps. f su. edu/pub/JMA-SST-Index/>), is a sliding average over five months of sea surface temperature anomalies in a portion (4'S-4'N, 15OoW-9O0W) of the Pacific Ocean. Data was gathered since 1949 but could be reconstructed in the past from 1868 on. A third one is elaborated by K. Wolter, at NOOA's Climate Diagnostic Center, as a compound of six different measures called ME1 for Multivariate ENSO Index. It covers the contemporary period since 1950, and can be found at the web address a (already considered in Exercise 4.6.6,p. 172). 1) This system is non-normal, and thus has no orthogonal eigen-basis. Find the eigenvalues and the eigen vectors of L and its adjoint Lt (for the canonical scalar product).

+

2) Consider the energy K = i(X; Xi) and determine the condition on a and b rendering K definite negative (monotonic stability). When this condition is not satisfied, identify the region of the ( X 1 , X z ) plane where K > 0 and the direction pmax= X2/X1 that maximize the initial growth of the energy. Examine in particular the case a 1 and decays when

IC(k)l < 1. For the explicit scheme, this gives:

The upper bound condition brings nothing, whereas the lower bound implies -1 < & x p ( r ) for stability. This leads to:

-1 N by adding zeroes for - M / 2 + 1 5 m 5 - N / 2 and N / 2 + 1 5 m 5 M / 2 (in the figure, all products in the band outside the interior square with side N cancel). Going back to the physical space one obtains an evaluation of v and w on a lattice with M regularly spaced points, finer than the original lattice. The product vw is then computed at the nodes of this lattice and next Fourier transformed. It remains to drop the Fourier coefficients outside interval [ - N / 2 1,N / 2 ] to get the unaliased spectrum of the product. The most dangerous non-zero term is with m' = m" = N / 2 . Its alias, m' + m'' - M , gets off the band if m' mr' - M < - N / 2 1, thus if 3 N / 2 - 1 < M , so that one can take M = 3 N / 2 ("312 rule"). Despite manipulations implied by the size increase from N to M , the evaluation of convolutions involved in the treatment of nonlinear terms is freed from a systematic error.

+

+

+

+

+

B. Numerical Approach

369

Fig. B.5 As in Figure B.4c, but now with M instead of N , the aliased terms belong to the gray-shaded triangular regions. In the regions with uniform shading, they arise from terms that are zero by construction (Iml outside the [ - N / 2 1,N / 2 ] interval). The nontrivial aliased terms belong to hatched regions. Some still belong to the domain bounded by the thick line when M < 3 N / 2 (left), which is no longer the case for A4 3 N / 2 (right).

+

>

B.3 B.3.1

Exercises

25.llncation and round-off errors

gv

Consider the differential equation = -v with initial condition v(0) = 1 and exact solution v ( t ) = exp(-t) and point out the role of the order of a numerical scheme on the precision of the numerical integration by comparing the exact result at t = 1, v = l / e , to approximations obtained with first-order iterations (B.3) and (B.5), second-order schemes (B.6), (B.7), (B.8), (B.9, B.lO) and (B.11, B.12). In particular, show formally that the RK schemes are of order At'. Write down all the corresponding programs and determine the distance ~ V N- l/el between the exact solution and the numerical solution V N obtained performing N successive iterations of each scheme with a time-interval At = 1/N. Use logarithmic scales to plot this distance as a function of At and to identify the regime dominated by truncation errors from that dominated by the accumulation of round-off errors. Test also the fourth-order Runge-Kutta scheme (B.13). B.3.2

Stability of multi-step schemes

Consider the integration of &v = f (v) by means of second-order schemes (B.7) and (B.8) and assume that a solution {v,,n = 0 , 1 , . . . } is known.

Instabilities, Chaos and ‘h-bulence

370

Neighboring solutions {V,, n = 0,1, . . .} can be written as series of perturbations {un, n = 0 , 1 , . . . }, with u, = V , - v, governed by the linearized map gn = df/dvln. Show that the two numerical schemes respectively lead to:

+

un+1= un +At ( 3 g n ~ n- gn-Ian-1) U,+I = un-1+ 2At gnu,.

7

(B.32) (B.33)

Then write (B.32) and (B.33) as two-dimensional maps for the two variables un and v, = u,-1 and compute their eigenvalues. Determine which is the physical eigenvalue Xphys controlling the solution t o the linearized problem $u = g(v)u, where v is the solution to the primitive problem $ v = f ( v ) , and which is the eigenvalue A, corresponding to the spurious numerical mode. The instability takes place when lAnuml > 1. Assuming that 191 is bounded by some gmaxfor the values of v of interest, discuss the stability properties of the two schemes. B.3.3

Finite difference schemes for the S H model

Derive second-order consistent finite difference approximations to the linear part of the Swift-Hohenberg (SH) model:

a,v = Tv -

(axx+ 1 ) 2 v 5

(T

- 1). - dxxxxv - 2dxxv.

(B.34)

Consider explicit-Euler, implicit-Euler and Crank-Nicholson temporal schemes and use (B.18) to discretize the fourth-order derivative. Study their stability by introducing discrete Fourier normal modes analogous t o those leading to conditions (B.23) and (B.24). Write down quasi-linearized Crank-Nicholson schemes for the original SH model, i.e. (B.34) with nonlinear term N ( v ) = -v3 added to its r.h.s. by expanding the nonlinear terms N(v) as:

Consider also the modified SH model completed by a term N(v) = -vdxv on the r.h.s. ( i e . an advection term vd,v on its 1.h.s).

B. Numerical Approach

B.4

Case

371

studies

The working sessions are organized around two themes: ODEs and PDEs. To deal with ODEs, we choose the simplest fourth-order Runge-Kutta scheme (B.13) and apply it to dynamical systems with few degrees of freedom, the periodically forced pendulum (two-dimensional, non-autonomous, sB.4.1) and the Lorenz model (three-dimensional, autonomous, fjB.4.2). Other examples are illustrated but not studied in detail, the Rossler system, jB.4.3.1, and the Chua circuit, sB.4.3.2. All this can serve to materialize concepts introduced in Chapters 2 and 4. Problems arising in EDPs are illustrated using two variants of the SH model introduced in some exercises of Chapters 3 and 4, and considered in exercise B.3.3 above. They may help to better understand pattern formation studied in Chapter 5. The approach can be extended to illustrate the generation of dissipative waves. Boundary conditions of Neumann-Dirichlet type are better adapted to finite difference methods and will mainly serve us t o illustrate numerical stability problems. Periodic boundary conditions are ides1 for a first approach of spectral methods used t o illustrate spacetime chaotic regimes. Computational Fluid Dynamics (CFD) will be left t o specialists. B.4.1

ODEs 1: Forced pendulum

We consider a pendulum submitted to a sinusoidal external force:

$e

+ q-&e + sin(e) = f cos(wt)

(B.35)

where B is the angle that the pendulum makes with the vertical, q is the damping coefficient, f is the intensity of the forcing and w its angular frequency. Write down a second-order Runge-Kutta program (B.11, B.12) for (B.35) written as a system of two first-order ODEs for 0 and 4 = 4, taking care of the explicit time dependence introduced by the forcing. Then turn to a fourth-order scheme (B.13). 1) Use that routine t o draw the projection of the trajectories in the plane (0,4). Choose for example w = f , r] = 0.5 and f variable in the range [0.5,3]. Consider in particular cases f = 1.07 and 1.15. See Figure B.6. 2) Perform a stroboscopic analysis (PoincarC section) of the system at the forcing period T = 21r/w and observe the attractor in particular for the previous parameter values mentioned. See Figure B.7, left.


372

70r

25r

2~ 1 5 ~ I~ 05-

0-

-

-0.5

1- 1 5 ~

2-2 5

0

10

20

30

I0

50

50

70

Fig. B.6 Forced pendulum for w = 2/3, f = 1.15. Top-left: Time series of 0. Top-right: Time series of 0 z 4. Bottom-left: q5 as a function of 0. Bottom-right: Three-dimensional representation in the space ( t , O , q $ ) , +axis vertical, &axis to the left, t-axis t o the right.

3) When the intensity of the forcing f increases, one finds parameter ranges where the attractor is periodic and other ranges where it is chaotic. Draw the bifurcation diagram obtained by varying f by recording the values reached by variables q5 on the surface of section (for every At = T), at given w E [ w l ,wz], for a sufficiently long trajectory and after having eliminated points corresponding to the transient. See Figure B.7, right. For all these simulations, it will be interesting to choose the time step At such that At = T / N so that a full oscillation of the forcing corresponds exactly to an integer number of time steps. N will be sufficiently large, so that one has At

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